\(C(\infty,\infty)\):  The Axiom of Choice: Every  set  of  non-empty sets has a choice function.

Existence of successor cardinals: For every cardinal \(m\) there is a cardinal \(n\) such that \(m < n\) and \((\forall p < n)(p \le m)\).

Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means \(\cong \aleph_0\).)

\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.

\(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable  family  of denumerable subsets of \({\Bbb R}\) is denumerable.

There is no infinite decreasing sequence of cardinals.

\(C(\aleph_{0},\infty)\):

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

\(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function.

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\), \(A\) has an infinite subset \(B\) such that for every \(n\in\omega\), \(n>0\), the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco [1998b]

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\) and every \(n\in\omega\), there is an infinite subset \(B\) of \(A\) such the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco} [1998b]

Every Dedekind finite subset of \({\Bbb R}\) is finite.

BPI: Every Boolean algebra has a prime ideal.

\(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every  set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)).  

\(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) has a choice function.

Ramsey's Theorem I: If \(A\) is an infinite set and the family of all 2 element subsets of \(A\) is partitioned into 2 sets \(X\) and \(Y\), then there is an infinite subset \(B\subseteq A\) such that all 2 element subsets of \(B\) belong to \(X\) or all 2 element subsets of \(B\) belong to \(Y\). (Also, see Form 325.), Jech [1973b], p 164 prob 11.20.

\(PUT(\aleph_{0},2,\aleph_{0})\):  The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.

A real function is analytically representable if and only if it is in Baire's classification. G.Moore [1982], equation (2.3.1).

If \(\{A_{x}: x \in S \}\) and \(\{B_{x}: x \in  S\}\) are families  of pairwise disjoint sets and \( |A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}| = |\bigcup_{x\in S} B_{x}|\). Moore [1982] (1.4.12 and 1.7.8).

If \(S\) is well ordered, \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families of pairwise disjoint sets, and \(|A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}|= |\bigcup_{x\in S} B_{x}|\). G\.

\(UT(2^{\aleph_{0}},2^{\aleph_{0}},2^{\aleph_{0}})\): If every member of an infinite  set of cardinality \(2^{\aleph _{0}}\) has power \(2^{\aleph_{0}}\), then the union has power \(2^{\aleph_{0}}\).

\((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\).

\(C(\aleph_0,2^{(2^{\aleph_0})})\): Every denumerable collection of non-empty sets each with power \(2^{(2^{\aleph_{0}})}\) has a choice function.

\(\aleph _{\beta +1}\) is regular for all ordinals \(\beta\).

\(UT(\aleph_{0},2^{\aleph_{0}},2^{\aleph_{0}})\): The union of denumerably many sets each of power \(2^{\aleph _{0}}\) has power \(2^{\aleph_{0}}\).

\((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The  union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). Moore, G. [1982], p 36.

(Where \(p\) is a prime) AL20(\(\mathbb Z_p\)): Every vector space \(V\) over \(\mathbb Z_p\) has the property that every linearly independent subset can be extended to a basis.  (\(\mathbb Z_p\) is the \(p\) element field.) Rubin, H./Rubin, J. [1985], p. 119, Statement AL20

Ordering Principle: Every set can be linearly ordered.

\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem:  The union of a denumerable set of denumerable sets is denumerable.

\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets  has a choice function.

If \(n\in\omega-\{0,1\}\), \(C(LO,n)\):  Every linearly ordered set of \(n\) element sets has  a choice function.

\(\aleph_{1}\) is regular.

The union of countably many meager subsets of \({\Bbb R}\) is meager. (Meager sets are the same as sets of the first category.) Jech [1973b] p 7 prob 1.7.

Compact T\(_2\) spaces are Loeb. (A space is Loeb if the set of non-empty closed sets has a choice function.)

Lebesgue measure is countably additive.

\({\Bbb R}\) is not the union of a countable family of countable sets.

\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

\(W_{\aleph _{1}}\): For every cardinal \(m\), \(m \le \aleph_{1}\) or \(\aleph_{1}\le m \).

Löwenheim-Skolem Theorem: If a countable family of first order  sentences is satisfiable in a set \(M\) then it is satisfiable in a countable subset of \(M\). (See Moore, G. [1982], p. 251 for references.

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

\(DC(\aleph _{1})\):  Given a relation \(R\) such that for every  subset \(Y\) of a set \(X\) with \(|Y| < \aleph_{1}\) there is an \(x \in  X\)  with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow  X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\).

If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function.

If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(\infty,K)\): For every \(n\in K\), every set of \(n\)-element sets has a choice function.

If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function.

If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(WO,K)\): For every \(n\in K,\) \(C(WO,n)\).

Order Extension Principle: Every partial ordering can be extended to a linear ordering.  Tarski [1924], p 78.

Sikorski's  Extension Theorem: Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). Sikorski [1964], p. 141.

Cofinality Principle: Every linear ordering has a cofinal sub well ordering.  Sierpi\'nski [1918], p 117.

Hahn-Banach Theorem:  If \(V\) is a real vector space and \(p: V \rightarrow {\Bbb R}\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t > 0)( p(tx) = tp(x) )\) and \(S\) is a subspace of \(V\) and \(f:S \rightarrow {\Bbb R}\) is linear and satisfies \((\forall  x \in S)( f(x) \le  p(x) )\) then \(f\) can be extended to \(f^{*} : V \rightarrow {\Bbb R}\) such that \(f^{*}\) is linear and \((\forall x \in V)(f^{*}(x) \le p(x))\).

For all infinite cardinals \(m\), \(m^2\le 2^m\).  Mathias [1979], prob 1336.

For all infinite cardinals \(m\), \(m\) adj \(2^{m}\) implies \(m\) is an aleph.  
Note that \(m\) adj \(n\) iff \(m < n\) and \(\neg (\exists p) (m < p < n).\)
Mathias [1979] prob 1337, thm 1338 and prob 1339.

For all infinite cardinals \(m\) and \(n\), if \(m\cdot n = m + n\) then, \(m \le  n\) or \(n \le m\).
Mathias [1979], p 125.

\(\aleph(2^{\aleph_{0}})\neq\aleph_{\omega}\). (\(\aleph(2^{\aleph_{0}})\) is Hartogs' aleph, the least \(\aleph\) not \(\le 2^{\aleph_{0}}\).)
Mathias [1979], p 125.

If \(x\) and \(y\) are Dedekind finite sets then either \(|x|\le |y|\) or \(|y|\le |x|\).
Mathias [1979], p 125.

There is an ordinal \(\alpha\) such that \(\aleph(2^{\aleph_{\alpha }})\neq\aleph_{\alpha +1}\). (\(\aleph(2^{\aleph_{\alpha}})\) is Hartogs' aleph, the least \(\aleph\) not \(\le 2^{\aleph _{\alpha}}\).)
Mathias [1979], p 126.

If \((A,\le)\) is a partial ordering that is not a well ordering, then there is no set \(B\) such that \((B,\le)\) (the usual injective cardinal ordering on \(B\)) is isomorphic to \((A,\le)\).
Mathias [1979], p 120.

\(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.
Moore, G. [1982], p 125.

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
Jech [1973b], p 172 prob 8.5.

\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)

The Krein-Milman Theorem: Let \(K\) be a compact convex set in a locally convex topological vector space \(X\). Then \(K\) has an extreme point. (An extreme point is a point which is not an interior point of any line segment which lies in  \(K\).) Rubin, H./Rubin, J. [1985] p. 177.

Every vector space over a field has a basis.

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

Nielsen-Schreier Theorem: Every subgroup of a free group is free.  Jech [1973b], p 12.

Every field has an algebraic closure.  Jech [1973b], p 13.

There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.

\(W_{\aleph_{\alpha}}\): \((\forall x)(|x|\le\aleph_{\alpha }\) or \(|x|\ge \aleph_{\alpha})\). Jech [1973b], page 119.

Artin-Schreier Theorem:  Every field in which \(-1\) is not the sum of squares can be ordered. (The ordering, \(\le \), must satisfy (a) \(a\le b\rightarrow a + c\le b + c\) for all \(c\) and (b) \(c\ge 0\) and \(a\le b\rightarrow a\cdot c\le b\cdot c\).)

\(\forall n\in\omega\), \(PC(\infty,n,\infty)\):  For every \(n\in\omega\), if \(C\) is an infinite family of \(n\) element sets, then \(C\) has an infinite subfamily with a choice function. De la Cruz/Di Prisco [1998b]

For every \(A\subseteq\Bbb R\) the following are equivalent:

1. \(A\) is closed and bounded.
2. Every sequence \(\{x_{n}\}\subseteq A\) has a convergent subsequence with limit in A.

If a set has at least two elements, then it can be partitioned into well orderable subsets, each of which has at least two elements.

\(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).

A linear ordering of a set \(P\) is a well ordering if and only if \(P\) has no infinite descending sequences. Jech [1973b], p 23.

Urysohn's Lemma:  If \(A\) and \(B\) are disjoint closed sets in a normal space \(S\), then there is a continuous \(f:S\rightarrow [0,1]\) which is 1 everywhere in \(A\) and 0 everywhere in \(B\). Urysohn [1925], pp 290-292.

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

(For \(n\in\omega\)) \(K(n)\): For every set \(S\) there is an ordinal \(\alpha\) and a one to one function \(f: S \rightarrow {\cal P}^{n}(\alpha)\). (\({\cal P}^{0}(X) = X\) and \({\cal P}^{n+1}(X) = {\cal P}({\cal P}^{n}(X))\). (\(K(0)\) is equivalent to [1 AC]  and \(K(1)\) is equivalent to the selection principle (Form 15)).

\(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.)

\(E(I,II)\) Howard/Yorke [1989]: \(T\)-finite is equivalent to finite.

\(E(II,III)\) (Howard/Yorke [1989]): \((\forall x)(x\) is \(T\)-finite  if and only if \(\cal P(x)\) is Dedekind finite).

\(C(\infty,\aleph_{0})\):  Every family of denumerable sets has  a choice function.  Jech [1973b] p 115 prob 7.13.

\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) has a choice function.

\(DC(\aleph_{\alpha})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(|Y|<\aleph_{\alpha}\), there is an \(x\in X\) with \(Y\mathrel R x\) then there is a function \(f:\aleph_{\alpha}\to X\) such that (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\).

  \(C(\infty ,2)\):  Every family of pairs has a choice function.

Antichain Principle:  Every partially ordered set has a maximal antichain. Jech [1973b], p 133.

\(LW\):  Every linearly ordered set can be well ordered. Jech [1973b], p 133.

\(PW\):  The power set of a well ordered set can be well ordered.

\(C(WO,{\Bbb R})\):  Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function.

There is a non-measurable subset of \({\Bbb R}\).

\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals  has a choice function. Jech [1973b], p 148 prob 10.1.

Existence of Complementary Subspaces over a Field \(F\): If \(F\) is a field, then every vector space \(V\) over \(F\) has the property that if \(S\subseteq V\) is a subspace of \(V\), then there is a subspace \(S'\subseteq V\) such that \(S\cap S'= \{0\}\) and \(S\cup S'\) generates \(V\). H. Rubin/J. Rubin [1985], pp 119ff, and Jech [1973b], p 148 prob 10.4.

Löwig's Theorem:If \(B_{1}\) and \(B_{2}\) are both bases for the vector space \(V\) then \(|B_{1}| = |B_{2}|\).

Cardinal Representatives: For every set \(A\) there is a function \(c\) with domain \({\cal P}(A)\) such that for all \(x, y\in {\cal P}(A)\), (i) \(c(x) = c(y) \leftrightarrow x\approx y\) and (ii) \(c(x)\approx x\).  Jech [1973b] p 154.

The set of all finite subsets of a Dedekind finite set is Dedekind finite. Jech [1973b] p 161 prob 11.5.

Rado's Selection Lemma: Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family  of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\).  Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S.

Weak Partition Principle:  For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\).

Partition Principle:  If \(S\) is a partition of \(M\), then \(S \precsim M\).

For all Dedekind finite cardinals \(p\) and \(q\), if \(p^{2} = q^{2}\) then \(p = q\). Jech [1973b], p 162 prob 11.12.

If \((P,<)\) is a linear ordering and \(|P| > \aleph_{1}\) then some initial segment of \(P\) is uncountable. Jech [1973b], p 164 prob 11.21.

There is a regular uncountable aleph. Jech [1966b], p 165 prob 11.26.

There is a  partially ordered set \((A,\le)\) such that for no set \(B\) is \((B,\le)\) (the ordering  on \(B\) is the usual injective cardinal ordering) isomorphic to \((A,\le)\).

Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.

M. Hall's Theorem: Let \(\{S(\alpha): \alpha\in A\}\) be a collection of finite subsets (of a set \(X\)) then if

There is an ordinal \(\alpha\) such that \(2^{\aleph _{\alpha}}\) is not the union of a denumerable set of denumerable sets.

Every field \(F\) and every vector space \(V\) over \(F\) has the property that each linearly independent set \(A\subseteq V\) can be extended to a basis. H.Rubin/J.~Rubin [1985], pp 119ff.

Every vector space over \(\Bbb Q\) has a basis.

\(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set.

\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).

Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact.

Every A-bounded \(T_2\) topological space is weakly Loeb. (\(A\)-bounded means amorphous subsets are relatively compact. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.)

The product of weakly Loeb \(T_2\) spaces is weakly Loeb. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.)

Every compact \(T_2\) space is weakly  Loeb. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.

If \(\kappa\) is a measurable cardinal, then \(\kappa\) is the \(\kappa\)th inaccessible cardinal.

Every linearly orderable topological space is normal.  Birkhoff [1967], p 241.

van Douwen's choice principle: \(C(\aleph_{0}\),uniformly orderable with order type of the integers): Suppose \(\{ A_{i}: i\in\omega\}\) is a set and there is a function \(f\) such that for each \(i\in\omega,\ f(i)\) is an ordering of \(A_{i}\) of type \(\omega^{*}+\omega\) (the usual ordering of the integers), then \(\{A_{i}: i\in\omega\}\) has a choice function.

If \(K\subseteq\omega-\{0,1\}\), \(C(LO,K)\): Every linearly ordered set of non-empty sets each of whose cardinality is in \(K\) has a choice function.

\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function.

\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.

\(SPI^*\): Uniform weak ultrafilter principle: For each family \(F\) of infinite sets \(\exists f\) such that \(\forall x\in F\), \(f(x)\) is a non-principal ultrafilter on \(x\).

Every operator on a Hilbert space with an amorphous base is the direct sum of a finite matrix and  a  scalar operator.  (A set is amorphous if it is not the union of two disjoint infinite sets.)

There does not exist an infinite, compact connected \(p\) space. (A \(p\) space is a \(T_2\) space in which the intersection of any well orderable family of open sets is open.)

\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).

An amorphous power of a compact \(T_2\) space, which as a set is well orderable, is well orderable.

Aczel's Realization Principle: On every infinite set there is a Hausdorff topology with an infinite set of non-isolated points.

For every infinite set \(A\), \(A\) admits a partition into sets of order type \(\omega^{*} + \omega\). (For every infinite \(A\), there is a set \(\{\langle C_j,<_j \rangle: j\in J\}\) such that \(\{C_j: j\in J\}\) is a partition of \(A\) and for each \(j\in J\), \(<_j\) is an ordering of \(C_j\) of type \(\omega^* + \omega\).)

\({\cal P}(\Bbb R)\) is well orderable.

\(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).

\(PC(\infty, <\aleph_0,\infty)\):  Every infinite family of finite  sets has an infinite subfamily with a choice function.

Every set is either well orderable or has an infinite amorphous subset.

If \(X\) is an infinite \(T_1\) space and \(X^{Y}\) is \(T_5\), then \(Y\) is countable. (\(T_5\) is 'hereditarily \(T_4\)'.)

If \(X\) is a \(T_2\) space with at least two points and \(X^{Y}\) is hereditarily metacompact then \(Y\) is  countable. (A space is metacompact if every open cover has an open point finite refinement. If \(B\) and \(B'\) are covers of a space \(X\), then \(B'\) is a refinement of \(B\) if \((\forall x\in B')(\exists y\in B)(x\subseteq y)\). \(B\) is point finite if \((\forall t\in X)\) there are only finitely many \(x\in B\) such that \(t\in x\).) van Douwen [1980]

Surjective Cardinal Cancellation (depends on \(k\in\omega-\{0\}\)): For all cardinals \(x\) and \(y\), \(kx\le^* ky\) implies \(x\le^* y\).

Suppose \(k\in\omega-\{0\}\). If \(f\) is a 1-1 map from \(k\times X\) into \(k\times Y\) then there are partitions \(X = \bigcup_{i \le k} X_{i} \) and \(Y = \bigcup_{i \le k} Y_{i} \) of \(X\) and \(Y\) such that \(f\) maps \(\bigcup_{i \le k} (\{i\} \times  X_{i})\) onto \(\bigcup_{i \le k} (\{i\} \times  Y_{i})\).

Suppose \(k\in\omega\).  If \(f\) is a partial map from \(k\times Y\)  onto \(k\times X\) (that is, the domain is a subset of \(k\times Y\)), then there are partitions \(X = \bigcup_{i \le k} X_{i}\)  and \(Y = \bigcup_{i \le k} Y_{i}\) of \(X\) and \(Y\) such that \(f\) maps \(\bigcup_{i \le k} (\{i\} \times Y_{i})\) onto \(\bigcup^{}_{i \le k} (\{i\} \times X_{i})\).

Using the discrete topology on 2, \(2^{\cal P(\omega)}\) is compact.

Let \(\Omega\) be the set of all (undirected) infinite cycles of reals (Graphs whose vertices are real numbers, connected, no loops and each vertex adjacent to  exactly two others). Then there is a function \(f\) on \(\Omega \) such that for all \(s\in\Omega\), \(f(s)\) is a direction along \(s\).

[14 P(\(n\))] with \(n = 2\):  Let \(\{A(i): i\in I\}\) be a collection of sets such that \(\forall i\in I,\ |A(i)|\le 2\) and suppose \(R\) is a symmetric binary relation on \(\bigcup^{}_{i\in I} A(i)\) such that for all finite \(W\subseteq I\) there is an \(R\) consistent choice function for \(\{A(i): i \in W\}\). Then there is an \(R\) consistent choice function for \(\{A(i): i\in I\}\).

\(\neg  PB\):  There is a set of reals without the property of Baire.  Jech [1973b], p. 7.

\(H(C,TR)\): If \((X,R)\) is a connected relation (\(u\neq v\rightarrow u\mathrel R v\) or \(v\mathrel R u\)) then \(X\) contains a \(\subseteq\)-maximal transitive subset.

Every set is almost well orderable.

Compact \(P_0\)-spaces are Dedekind finite. (A \(P_0\)-space is a topological space in which the intersection of a countable collection of open sets is open.)

\(A(F,A1)\): For every \(T_2\) topological space \((X,T)\), if \(X\) is a continuous finite to one image of an A1 space then \((X,T)\) is  an A1 space. (\((X,T)\) is A1 means if \(U \subseteq  T\) covers \(X\) then \(\exists f : X\rightarrow U\) such that \((\forall x\in X) (x\in f(x)).)\)

\(A(D2)\):  Every \(T_2\) topological space \((X,T)\) can be covered by a well ordered family of discrete sets.

\(A(W2,B1)\): For every \(T_2\) topological space \((X,T)\), if \(X\) is well ordered, then \(X\) has a well ordered base.

\(A(F)\):  Every \(T_2\) topological space is a continuous, finite to one image of an \(A1\) space.

\(PC(\infty,\aleph_0,\infty)\): Every infinite set of denumerable sets has an infinite subset with a choice function.

\(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well  orderable. (If \(\kappa\) is a well ordered cardinal, see note 27 for \(UT(WO,\kappa,WO)\).)

\(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets.  (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.)

The closed unit ball of a Hilbert space is compact in the weak topology.

Tychonoff's Compactness Theorem for Countably Many \(T_2\) Spaces: The product of countably many \(T_2\) compact spaces is compact.

Theorem of Gelfand and Kolmogoroff: Two compact \(T_2\) spaces are  homeomorphic if their rings of real valued continuous functions are isomorphic.

Theorem of Goodner: A compact \(T_{2}\) space is extremally disconnected (the closure of every open set is open) if and only if each non-empty subset of \(C(X)\) (set of continuous real valued functions on \(X\)) which is pointwise bounded has a supremum.

In every Hilbert space \(H\), if the closed unit ball is sequentially compact, then \(H\) has an orthonormal basis.

The regular cardinals are cofinal in the class of ordinals.

No Dedekind finite set can be mapped onto an aleph. (See [9 E].)

Definability of cardinal addition in terms of \(\le\): There is a first order formula whose only non-logical symbol is \( \le \) (for cardinals) that defines cardinal addition.

Non-existence of infinite units: There is no infinite cardinal number \(A\) such that \(A + A > A\)  and for all cardinals \(x\) and \(y\), \(x + y = A\rightarrow x = A\) or \(y = A\).

Every non-well-orderable set has an infinite, Dedekind finite subset.

Every non-well-orderable set has an infinite subset with a Dedekind finite power set.

\(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function.

\(PC(\infty,2,\infty)\): Every infinite family of pairs has an infinite subfamily with a choice function.

\(PKW(\aleph_{0},\ge 2,\infty)\), Partial Kinna-Wagner Principle:  For every denumerable family \(F\) such that for all \(x\in F\), \(|x|\ge 2\), there is an infinite subset \(H\subseteq F\) and a function \(f\) such that for all \(x\in H\), \(\emptyset\neq f(x) \subsetneq x\).

Dual Cantor-Bernstein Theorem:\((\forall x) (\forall y)(|x| \le^*|y|\) and \(|y|\le^* |x|\) implies  \(|x| = |y|)\) .

There is an uncountable subset of \({\Bbb R}\) without a perfect subset.

\(\aleph_{1}\le 2^{\aleph_{0}}\).

If \((P,\le)\) is a partial order such that \(P\) is the denumerable union of finite sets and all antichains in \(P\) are finite then for each denumerable family \({\cal D}\) of dense sets there is a \({\cal D}\) generic filter.

For every infinite set \(S\), if \(S\) is hereditarily countable  (that is, every \(y\in TC(S)\) is countable) then \(|TC(S)|= \aleph_{0}\).

\(MPL\): Metric spaces are para-Lindelöf.

\(RM1,\aleph_{\alpha }\): The representation theorem for multi-algebras with \(\aleph_{\alpha }\) unary operations:  Assume \((A,F)\) is  a  multi-algebra  with \(\aleph_{\alpha }\) unary operations (and no other operations). Then  there  is  an  algebra \((B,G)\)  with \(\aleph_{\alpha }\) unary operations and an equivalence relation \(E\) on \(B\) such that \((B/E,G/E)\) and \((A,F)\) are isomorphic multi-algebras.

Transitivity Condition: For all sets \(x\), there is a set \(u\) and a function \(f\) such that \(u\) is transitive and \(f\) is a one to one function from \(x\) onto \(u\).

Every infinite, locally finite group has an infinite Abelian subgroup. (Locally finite means every finite subset generates a finite subgroup.)

An infinite box product of regular \(T_1\) spaces, each of cardinality greater than 1, is neither first countable nor connected.

If  \(n\in\omega\), \(n\ge 2\) and \(N \subseteq \{ 1, 2, \ldots , n-1 \}\), \(N \neq\emptyset\), \(MC(\infty,n, N)\):  If \(X\) is any set of \(n\)-element sets then  there is  a function \(f\) with domain \(X\) such that for all \(A\in X\), \(f(A)\subseteq A\) and \(|f(A)|\in N\).

Suppose  \(\epsilon > 0\) is an ordinal. \(\forall x\), \(x\in W(\epsilon\)).

Every Abelian group has a divisible hull.  (If \(A\) and \(B\)  are groups, \(B\) is a divisible hull of \(A\) means \(B\) is a divisible group, \(A\) is a subgroup of \(B\) and for every non-zero \(b \in  B\), \(\exists  n \in \omega \)  such that \(0\neq nb\in A\).)  Fuchs [1970], Theorem 24.4 p 107.

\(C(2^{\aleph_0},\infty)\): Every set \(X\) of non-empty sets such that \(|X|=2^{\aleph_0}\) has a choice function.

There is an aleph whose cofinality is greater than \(\aleph_{0}\).

There are no \(\aleph_{\alpha}\) minimal  sets.  That is, there are no sets \(X\) such that

1. \(|X|\) is incomparable with \(\aleph_{\alpha}\)
2. \(\aleph_{\beta}<|X|\) for every \(\beta < \alpha \) and
3. \(\forall Y\subseteq X, |Y|<\aleph_{\alpha}\) or \(|X-Y| <\aleph_{\alpha}\).

Existence of a double uniformization: For all \(X\) and \(Y\), for all \(E\subseteq X\times Y\), if there is an infinite cardinal \(\kappa\) satisfying:

1. \(\forall y_0\in Y\), \(|\{(x,y_0): x\in X\}\cap E| =\kappa\) and
2. \(\forall x_0\in X\), \(|\{(x_0,y): y\in Y\}\cap E| = \kappa\),

Every linearly ordered Dedekind finite set is finite.

Every pair of cardinal numbers has a least upper bound (in the usual cardinal ordering.)

Every pair of cardinal numbers has a greatest lower bound (in the usual cardinal ordering.)

\(EP\ Ab\): For every Abelian group \(A\) there is a projective Abelian group \(G\) and a homomorphism from \(G\) onto \(A\).

\(EI\ Ab\): For every Abelian group \(A\) there is an injective Abelian group \(G\) and a one to one homomorphism from \(A\) into \(G\).

There is a non-trivial injective Abelian group.

\(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\).

\(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\).

\(EFP\ Ab\): Every Abelian group is a homomorphic image of a free projective Abelian group.

\(C(\varPi^1_2)\) or \(AC(\varPi^1_2)\): If \(P\in \omega\times{}^{\omega}\omega\), \(P\) has domain \(\omega\), and \(P\) is in \(\varPi^1_2\), then there is a sequence of elements \(\langle x_{k}: k\in\omega\rangle\) of \({}^{\omega}\omega\) with \(\langle k,x_{k}\rangle \in P\) for all \(k\in\omega\). Kanovei [1979].

Every general linear system has a linear global reaction.

\(\aleph_{\alpha}\) and \(\aleph_{\alpha+1}\) are not both measurable.

\({\Bbb R}^{2}\) is the union of three sets \(C\) with the property that for all \(x\in C\) there is a straight line \(L\) such that \(L\cap C = \{x\}\).

For every set \(S\), if the only linearly orderable subsets of \(S\) are the finite subsets of \(S\), then either \(S\) is finite or \(S\) has an amorphous subset.

(For \(n\in\omega-\{0,1\}\)) If all \(\varSigma^{1}_{n}\), Dedekind finite subsets of \({}^{\omega }\omega\) are finite, then all \(\varPi^1_n\) Dedekind finite subsets of \({}^{\omega} \omega\) are finite.

For all infinite \(x\), \(|2^{x}| = |x!|\).

Linking Axiom for Boolean Algebras: Every Boolean algebra has a maximal linked system. (\(L\subseteq B\) is linked if \(a\wedge b\neq 0\) for all \(a\) and \(b \in L\).)

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

\(C\)(disjoint,\(\subseteq\Bbb R)\): Every partition of \({\cal P}(\omega)\) into non-empty subsets has a choice function.

For every infinite \(X\), there is a function from \(X\) onto \(2X\).

For all cardinals \(m\) and \(n\), if \(m\le^* n\) and \(\neg (n\le^* m)\) then there is a cardinal \(k \le n\) such that \(m\le^* k\).

The existence of a non-principal ultrafilter: There exists an infinite set \(X\) and a non-principal ultrafilter on \(X\).

\(UT(\aleph_{\alpha },\aleph_{\alpha}, <2^{\aleph_{\alpha }})\): The union of \(\aleph_{\alpha}\) sets each of cardinality \(\aleph_{\alpha}\) has cardinality less than \(2^{\aleph_{\alpha}}\).

For all ordinals \(\alpha\), \(\aleph_{\alpha+1}\le 2^{\aleph_\alpha}\).

There is an ordinal \(\alpha\) such that for all \(X\), if \(X\) is a denumerable union of denumerable sets then \({\cal P}(X)\) cannot be partitioned into \(\aleph_{\alpha}\) non-empty sets.

The commutator subgroup of a free group is free.

\(DCR\): Dependent choice for relations on \(\Bbb R\): If \(R\subseteq\Bbb R\times\Bbb R\) satisfies \((\forall x\in \Bbb R)(\exists y\in\Bbb R)(x\mathrel R y)\) then there is a sequence \(\langle x(n): n\in\omega\rangle\) of real numbers such that \((\forall n\in\omega)(x(n)\mathrel R x(n+1))\).

\(C(2^{\aleph_{0}},\subseteq{\Bbb R})\): If \(R\) is a relation on \({\Bbb R}\) such that for all \(x\in{\Bbb R}\), there is a \(y\in{\Bbb R}\) such that \(x\mathrel R y\), then there is a function \(f: {\Bbb R} \rightarrow{\Bbb R}\) such that for all \(x\in{\Bbb R}\), \(x\mathrel R f(x)\).

\(C(\infty,\aleph_{1})\): If \((\forall y\in X)(|y| = \aleph_{1})\) then \(X\) has a choice function.

\(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(|f(y)|=\aleph_{0}\).

If \((\forall y\subseteq X)(y\) can be linearly ordered implies \(y\) is finite), then \(X\) is finite.

Every infinite tree has either an infinite chain or an infinite antichain.

Every infinite partially ordered set has either an infinite chain or an infinite antichain.

\((\forall n\in\omega - \{0\}) MC(\infty,\infty \), relatively prime to \(n\)): \(\forall n\in\omega -\{0\}\), if \(X\) is a set of non-empty sets, then  there  is  a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\) and \(|f(x)|\) is relatively prime to \(n\).

\((\forall n\in\omega-\{0\}) MC(\infty,WO\), relatively prime to \(n\)): For all non-zero \(n\in \omega\), if \(X\) is a set of non-empty well orderable sets, then there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\), and \(|f(x)|\) is relatively prime to \(n\).

Suppose \(p\in\omega\) and \(p\) is a prime. Any two elementary Abelian \(p\)-groups (all non-trivial elements have order \(p\)) of the same cardinality are isomorphic.

For all infinite \(X\), there is a non-principal measure on \(\cal P(X)\).

There is a non-principal measure on \(\cal P(\omega)\).

There is an infinite set \(X\) and a non-principal measure on \(\cal P(X)\).

There is a partition of the real line into \(\aleph_1\) Borel sets \(\{B_\alpha: \alpha<\aleph_1\}\) such that for some \(\beta <\aleph_1\), \(\forall\alpha <\aleph_1\), \(B_{\alpha}\in G_{\beta}\). (\(G_\beta\) for \(\beta < \aleph_1\) is defined by induction, \(G_0=\{A: A\) is an open subset of \({\Bbb R}\}\) and for \(\beta > 0\),

• \(G_\beta =\left\{\bigcup^\infty_{i=0}A_{i}: (\forall i\in\omega) (\exists\xi <\beta)(A_i\in G_\xi)\,\right\}\) if \(\beta\) is even and
• \(G_\beta = \left\{\bigcap^\infty_{i=0}A_{i}: (\forall i\in\omega) (\exists \xi < \beta)(A_{i}\in G_\xi)\,\right\}\) if \(\beta\) is odd.)

Every proper filter on \(\omega\) can be extended to an ultrafilter.

Let \(R\) be a commutative ring with identity, \(B\) a proper subring containing 1 and \(q\) a prime ideal in \(B\). Then there is a subring \(A\) of \(R\) and a prime ideal \(p\) in \(A\) such that

1. \(B\subseteq A\)
2. \(q = B\cap p\)
3. \(R - p\) is multiplicatively closed and
4. if \(A\neq R\),

For all groups \(G\), if every finite subgroup of \(G\) can be fully ordered then \(G\) can be fully ordered.

Every torsion free Abelian group can be fully ordered.

If \((G,\circ,\le)\) is a partially ordered group, then \(\le\) can be extended to a linear order on \(G\) if and only if for every finite set \(\{a_{1},\ldots, a_{n}\}\subseteq G\), with \(a_{i}\neq\) the identity for \(i = 1\) to \(n\), the signs \(\epsilon_{1}, \ldots,\epsilon_{n}\) (\(\epsilon_{i} = \pm 1\)) can be chosen so that \(P\cap S(a^{\epsilon_{1}}_{1},\ldots,a^{\epsilon_{n}}_{n})=\emptyset\) (where \(S(b_{1},\ldots,b_{n})\) is the normal sub-semi-group of \(G\) generated by \(b_{1},\ldots, b_{n}\) and \(P = \{g\in G: e\le g\}\) where \(e\) is the identity of \(G\).)

\(L^{1} = HOD\).

\(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable.

Every metric space \((X,d)\) has a \(\sigma\)-point finite base.

Artin-Schreier theorem: If a field has an algebraic closure it is unique up to isomorphism.

There is a non-Ramsey set: There is a set \(A\) of infinite subsets of \(\omega\) such that for every infinite subset \(N\) of \(\omega\), \(N\) has a subset which is in \(A\) and a subset which is not in \(A\).

If \(V\) is a vector space and \(B_{1}\) and \(B_{2}\) are bases for \(V\) then \(|B_{1}|\) and \(|B_{2}|\) are comparable.

If \(V\) is a vector space with a basis and \(S\) is a linearly independent subset of \(V\) such that no proper extension of \(S\) is a basis for \(V\), then \(S\) is a basis for \(V\).

The order of any group is divisible by the order of any of its subgroups, (i.e., if \(H\) is a subgroup of \(G\) then there is a set \(A\) such that \(|H\times A| = |G|\).)

Every elementary Abelian group (that is, for some prime \(p\) every non identity element has order \(p\)) is the direct sum of cyclic subgroups.

AL20(\(\mathbb Q\)):  Every vector \(V\) space over \(\mathbb Q\) has the property that every linearly independent subset of \(V\) can be extended to a basis. Rubin, H./Rubin, J. [1985], p.119, AL20.

If a group \(G\) satisfies "every ascending chain of subgroups is finite", then every subgroup of \(G\) is finitely generated.

Every algebraic closure of \(\Bbb Q\) has a real closed subfield.

There is, up to an isomorphism, at most one algebraic closure of \({\Bbb Q}\).

Every  principal ideal domain is a unique factorization domain.

Every principal ideal domain has a maximal ideal.

There is a function \(f :\omega_1\rightarrow \omega^{\omega}_1\) such that for every \(\alpha\), \(0 < \alpha < \omega_1\), \(f(\alpha )\) is a function from \(\omega\) onto \(\alpha\).

The monadic theory theory \(MT(\omega_1,<)\) of \(\omega_1\) is recursive.  \ac{Litman} \cite{1976} and note 85.

Every atomless Boolean algebra is Dedekind infinite.  \ac{Plotkin} \cite{1976}, notes 86 and 94.

For any \(\kappa\), \(\kappa\) is the cardinal number of an infinite complete Boolean algebra if and only if \(\kappa^{\aleph_0} = \kappa \).

If \(T\) is an infinite tree in which every element has exactly 2 immediate successors then \(T\) has an infinite branch.

\((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family of \(n\) element sets has a choice function.

The additive groups \(({\Bbb R},+)\) and \(({\Bbb C},+)\) are  isomorphic.

The additive groups of \(\Bbb Q\oplus\Bbb R\) and \(\Bbb R\) are isomorphic.

\L o\'s' Theorem: If \(M=\langle A,R_j\rangle_{j\in J}\) is a relational system, \(X\) any set and \({\cal F}\) an ultrafilter in \({\cal P}(X)\), then \(M\) and \(M^{X}/{\cal F}\) are elementarily equivalent.

\(Z(D,R,l)\): Every directed relation \((P,R)\) in which ramified subsets have least upper bounds, has a maximal element.

\(Z(D,R)\): Every directed relation \((P,R)\) in which every ramified subset \(A\) has an upper bound, has a maximal element.

\(Z(P,F)\): Every partially ordered set \((X,R)\) in which every forest \(A\) has an upper bound, has a maximal element.

\(Z(TR,P)\): Every transitive relation \((X,R)\) in which  every partially ordered subset has an upper bound, has a maximal element.

\(Z(D,L)\): Every directed relation \((X,R)\) in which linearly ordered subsets have upper bounds, has a maximal element.

\(Z(TR\&C,W)\): If \((X,R)\) is a transitive and connected relation in which every well ordered subset has an upper bound, then \((X,R)\) has a maximal element.

\(Z(TR\&C,P)\): If \((X,R)\) is a transitive and connected relation in which every partially ordered subset has an upper bound, then \((X,R)\) has a maximal element.

\(Z(TR,T)\): Every transitive relation \((X,R)\) in which every subset which is a tree has an upper bound, has a maximal element.

\(Z(TR,R)\): Every transitive relation \((X,R)\) in which every ramified subset \(A\) has an upper bound, has a maximal element.

\(H(AS\&C,P)\): Every every relation \((X,R)\) which is antisymmetric and connected contains a \(\subseteq\)-maximal partially ordered subset.

\(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set.

\(H(A,TR)\): Every relation \((X,R)\) contains a \(\subseteq\)-maximal transitive subset.

\(H(AS,P)\): Every antisymmetric relation contains \(\subseteq\)-maximal partially ordered subset.

There is no infinite, free complete Boolean algebra.

If \({\cal L}\)  is  a  lattice  isomorphic  to the  lattice  of subalgebras of some unary universal algebra (a unary universal algebra is one with only unary or nullary operations) and \(\alpha \) is an automorphism of \({\cal L}\) of order 2 (that is, \(\alpha ^{2}\)  is  the  identity) then there is a unary algebra \(\frak A\)  and an isomorphism \(\rho \) from \({\cal L}\) onto the lattice of subalgebras of \(\frak A^{2}\) with \[\rho(\alpha(x))=(\rho(x))^{-1} (= \{(s,t) : (t,s)\in\rho(x)\})\] for all \(x\in  {\cal L}\).

For every cardinal \(m\), there is a set \(A\) such that \(2^{|A|^2}\ge m\) and there is a choice function on the collection of 2-element subsets of \(A\).

\(CT_{\hbox{fin}}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas.

If \(n\in\omega-\{0,1\}\), \(CT_{n}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs in at most \(n\) formulas.

There is an \(X\subseteq{\Bbb R}\) such that neither \(X\) nor \(\Bbb R - X\) has a perfect subset.

There is a subset of \({\Bbb R}\) which is not Borel.

There is a cardinal number \(x\) and an \(n\in\omega\) such that \(\neg(x\) adj\(^n\, x^2)\). (The expression ``\(x\) adj\(^n\, ya\)" means there are cardinals \(z_0,\ldots, z_n\) such that \(z_0 = x\) and \(z_n = y\) and for all \(i,\ 0\le i < n,\ z_i< z_{i+1}\) and if  \(z_i < z\le z_{i+1}\), then \(z = z_{i+1}.)\) (Compare with [0 A]).

The sequence of cardinals \(\langle\aleph_n: n \in\omega\rangle\) has a unique minimal upper bound.

\(E(V'',III)\): For every set \(A\), \({\cal P}(A)\) is Dedekind finite if and only if \(A = \emptyset\)  or \(2|{\cal P}(A)| > |{\cal P}(A)|\). \ac{Howard/Spi\u siak} \cite{1994}.

\(E(D,VII)\): Every non-well-orderable cardinal is decomposable.

In an integral domain \(R\), if every ideal is finitely generated then \(R\) has a maximal proper ideal. note 45 E.

The Closed Graph Theorem for operations between Fréchet Spaces: Suppose \(X\) and \(Y\) are Fréchet spaces, \(T:X\to Y\) is linear and \(G=\{(x,Tx): x \in X \}\) is closed in \(X\times Y\). Then \(T\) is continuous. Rudin [1991] p. 51.

There is a complete separable metric space with a subset which does not have the Baire property.

There is a Hilbert space \(H\) and an  unbounded linear operator on \(H\).

\(\omega\not\to(\omega)^{\omega}\).

Cardinality of well ordered subsets:  For all \(n\in\omega\) and for all infinite \(x\), \(|x^n| < |s(x)|\) where \(s(x)\) is the set of all well orderable subsets of \(x\).

A system of linear equations over a field \(F\) has a solution in \(F\) if and only if every finite sub-system has a solution in \(F\).

Let \(E\) be a set and \(f: E\to E\), then \(f\) has a fixed point if and only if \(E\) is not the union of three mutually disjoint sets \(E_1\), \(E_2\) and \(E_3\) such that \(E_i \cap f(E_i) = \emptyset\) for \(i=1, 2, 3\).

Extended Krein-Milman Theorem:  Let K be a quasicompact (sometimes called convex-compact), convex subset of a locally convex topological vector space, then K has an extreme point. H. Rubin/J. Rubin [1985], p. 177-178.

The Hahn-Banach Theorem for Separable Normed Linear Spaces:  Assume \(V\) is a separable normed linear space and \(p :V \to \Bbb R\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t \ge 0)(\forall x \in V)(p(tx) = tp(x))\) and assume \(f\) is a linear function from a subspace \(S\) of \(V\) into \(\Bbb R\) which satisfies \((\forall x \in S)(f(x) \le p(x))\), then \(f\) can be extended to \(f^* :V \to \Bbb R\) so that \(f^* \) is linear and \((\forall x \in V)(f^*(x) \le p(x))\).

If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function.

If \(S\) is a set of subsets of a countable set and \(S\) is closed under chain unions, then \(S\) has a \(\subseteq\)-maximal element.

For all infinite \(x\), \(|2^x|=|x^x|\).

For all infinite \(x\), \(|x!|=|x^x|\).

\(MC(LO,\infty)\): For each linearly ordered family of non-empty sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is non-empty, finite subset of \(x\).

For all sets \(x\) and \(y\), if \(x\) can be linearly ordered and there is a mapping of \(x\) onto \(y\), then \(y\) can be linearly ordered.

Every linearly ordered \(W\)-set is well orderable.

DO:  Every infinite set has a dense linear ordering.

Part-\(\infty\): Every infinite set is the disjoint union of infinitely many infinite sets.

Extremally disconnected compact Hausdorff spaces are projective in the category of all compact Hausdorff spaces.

Every compact Hausdorff space has a Gleason cover.

Any extremally disconnected compact Hausdorff space is projective in the category of Boolean topological spaces.

Any continuous surjection between extremally disconnected compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain.

Any continuous surjection between Boolean spaces has an irreducible restriction to a closed subset of its domain.

Any continuous surjection between compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain.

If \(B\) is a Boolean algebra, \(S\subseteq B\) and \(S\) is closed under \(\land\), then there is a \(\subseteq\)-maximal proper ideal \(I\) of \(B\) such that \(I\cap S= \{0\}\).

There does not exist a \(T_2\) topological space \(X\) such that every infinite subset of \(X\) contains an infinite compact subset.

There are \(2^{\aleph_0}\) Vitali equivalence classes. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in{\Bbb Q})(x-y=q)\).). \ac{Kanovei} \cite{1991}.

The set of Vitali equivalence classes is linearly orderable. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow (\exists q\in{\Bbb Q})(x-y = q)\).).

If \(m\) is the cardinality of the set of Vitali equivalence classes, then \(H(m) = H(2^{\aleph_0})\), where \(H\) is Hartogs aleph function and the {\it Vitali equivalence classes} are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in {\Bbb Q})(x-y=q)\).

If \(p\) is a prime and if \(\{G_y: y\in Y\}\) is a set of finite groups, then the weak direct product \(\prod_{y\in Y}G_y\) has a maximal \(p\)-subgroup.

The Banach-Tarski Paradox: There are three finite partitions \(\{P_1,\ldots\), \(P_n\}\), \(\{Q_1,\ldots,Q_r\}\) and \(\{S_1,\ldots,S_n, T_1,\ldots,T_r\}\) of \(B^3 = \{x\in {\Bbb R}^3 : |x| \le 1\}\) such that \(P_i\) is congruent to \(S_i\) for \(1\le i\le n\) and \(Q_i\) is congruent to \(T_i\) for \(1\le i\le r\).

The Measure Extension Theorem: Suppose that \(\cal A_0\) is a subring (that is, \(a,b \in \cal A_0  \to a\vee b \in \cal A_0\) and \(a-b \in \cal A_0\)) of a Boolean algebra \(\cal A\) and \(\mu\) is a measure on \(\cal A_0\) (that is, \(\mu:\cal A \to [0,\infty]\), \(\mu(a\vee b) =\mu(a)+\mu(b)\) for \(a\land b = 0\), and \(\mu(0) = 0\).) then there is a measure on \(\cal A\) that extends \(\mu\).

Abelian groups are amenable. (\(G\) is amenable if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G)=1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).)

A subgroup of an amenable group is amenable. (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g \in G\), \(\mu(gA)=\mu(A)\).)

\(\Bbb Z\) (the set of integers under addition) is amenable.  (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).)

For every set \(X\) and every permutation \(\pi\) on \(X\) there are two reflections \(\rho\) and \(\sigma\) on \(X\) such that \(\pi =\rho\circ\sigma\) and for every \(Y\subseteq X\) if \(\pi[Y]=Y\) then \(\rho[Y]=Y\) and \(\sigma[Y]=Y\).  (A reflection is a permutation \(\phi\) such that \(\phi^2\) is the identity.) \ac{Degen} \cite{1988}, \cite{2000}.

\(\Omega = \omega_1\), where
\(\Omega = \{\alpha\in\hbox{ On}: (\forall\beta\le\alpha)(\beta=0 \vee (\exists\gamma)(\beta=\gamma+1) \vee\)
there is a sequence \(\langle\gamma_n: n\in\omega\rangle\) such that for each \(n\),
\(\gamma_n<\beta\hbox{ and } \beta=\bigcup_{n<\omega}\gamma_n.)\} \)

If a linearly ordered set \((A,\le)\) has the fixed point property then \((A,\le)\) is complete. (\((A,\le)\)  has the fixed point property if every function \(f:A\to A\) satisfying \((x\le y \Rightarrow f(x)\le f(y))\) has a fixed point, and (\((A,\le)\) is complete if every subset of \(A\) has a least upper bound.)

Weak Sikorski Theorem:  If \(B\) is a complete, well orderable Boolean algebra and \(f\) is a homomorphism of the Boolean algebra \(A'\) into \(B\) where \(A'\) is a subalgebra of the Boolean algebra \(A\), then \(f\) can be extended to a homomorphism of \(A\) into \(B\).

\(\aleph_1\) is not measurable.

Measurable cardinals are inaccessible.

No successor cardinal, \(\aleph_{\alpha+1}\), is measurable.

There does not exist an ordinal \(\alpha\) such that \(\aleph_{\alpha}\) is weakly compact and \(\aleph_{\alpha+1}\) is measurable.

\(KW(WO,\infty)\), The Kinna-Wagner Selection Principle for a well ordered family of sets: For every  well ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).  (See Form 15).

\(KW(\infty,WO)\), The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set \(M\) of well orderable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\).  (See Form 15.)

\(KW(WO,WO)\), The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets: For every well ordered set \(M\) of well orderable sets, there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

Ramsey's Theorem II: \(\forall n,m\in\omega\), if A is an infinite set and the family of all \(m\) element subsets of \(A\) is partitioned into \(n\) sets \(S_{j}, 1\le j\le n\), then there is an infinite subset \(B\subseteq A\) such that all \(m\) element subsets of \(B\) belong to the same \(S_{j}\). (Also, see Form 17.)

2-SAT:  Restricted Compactness Theorem for Propositional Logic III:   If \(\Sigma\) is a set of formulas in a propositional language such that every finite subset of \(\Sigma\) is satisfiable and if every formula in \(\Sigma\) is a disjunction of at most two literals, then \(\Sigma\) is satisfiable. (A literal is a propositional variable or its negation.) Wojtylak [1999] (listed as Wojtylak [1995])

\(KW(WO,<\aleph_0)\),  The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\)  then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

\(MC(WO,\infty)\): For  every well ordered set \(X\) such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that and for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.)

\(MC(\infty,WO)\): For  every set \(M\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).  (See Form 67.)

\(MC(WO,WO)\): For every well ordered set \(X\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).  (See Form 67.)

If \((X_i)_{i\in I}\) is a family of compact non-empty topological spaces then there is a family \((F_i)_{i\in I}\) such that \(\forall i\in I\), \(F_i\) is an irreducible closed subset of \(X_i\).

A product of non-empty compact sober topological spaces is non-empty.

\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd.

\(MC(\infty,\infty,\hbox{ even})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 2\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is even.

Every quotient group of an Abelian group each of whose elements has order  \(\le n\) has a set of representatives.

(For \(n\in\omega\), \(n\ge 2\).)  For every infinite set \(X\), there is an infinite \(Y \subseteq X\) such that the set of all \(n\)-element subsets of \(Y\) has a choice function.

\(C(WO\), uniformly linearly ordered):  If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\).

\(UT(\aleph_0,\aleph_0,WO)\): The union of a denumerable number of denumerable sets is well orderable.

Martin's Axiom \((\aleph_{0})\): Whenever \((P\le)\) is a non-empty, ccc  quasi-order (ccc means every anti-chain is countable) and \({\cal D}\) is a family of \(\le\aleph_0\) dense subsets of \(P\), then there is a \({\cal D}\) generic filter \(G\) in \(P\).

Every Lindelöf metric space is separable.

Every Lindelöf metric space is second countable.

(For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\):  Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See Form 166.)

A product of non-empty, compact \(T_2\) topological spaces is non-empty.

If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\).

Rasiowa-Sikorski Axiom:  If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\).

If \(V\) is a vector space without a finite basis then \(V\) contains an infinite, well ordered, linearly independent subset.

Idemmultiple Partition Principle: If \(y\) is idemmultiple (\(2\times y\approx y\)) and \(x\precsim ^* y\), then \(x\precsim y\).

If \(G\) is a group and \(X\) and \(Y\) both freely generate \(G\) then \(|X| = |Y|\).

\(MC(\infty,\aleph_0)\): For every set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).

\(MC(\aleph_0,\aleph_0)\): For every denumerable set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).

A countable product of metrizable spaces is metrizable.

A countable product of second countable spaces is second countable.

A countable product of first countable spaces is first countable.

A countable product of separable \(T_2\) spaces is separable.

\(KW(\aleph_0,\infty)\), The Kinna-Wagner Selection Principle for a denumerable family of sets: For every denumerable set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\).

  \(KW(\infty,\aleph_0)\), The Kinna-Wagner Selection Principle for a family of denumerable sets: For every set \(M\) of denumerable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\)  then \(\emptyset\neq f(A)\subsetneq A\).

\(KW(\aleph_0,\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of denumerable sets: For every denumerable set \(M\) of denumerable sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\).

\(KW(\aleph_0,<\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\).

If \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families  of pairwise disjoint sets and \( |A_{x}| \le |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}| \le |\bigcup_{x\in S} B_{x}|\).

A system of linear equations over the field \(\{0,1\}\) has a solution, if and only if every finite subsystem has a solution. Brunner [2001]

In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325.

In \(\Bbb R\), every Borel set is analytic. G. Moore [1982], pp 181 and 325.

There are exactly \(2^{\aleph_0}\) Borel sets in \(\Bbb R\). G. Moore [1982], p 325.

In \(\Bbb R\), there is a measurable set that is not Borel.  G. Moore [1982], p 325.

For every uncountable set \(A\), if \(A\) has the same cardinality as each of its uncountable subsets then \(|A| = \aleph_1\).

There is a discontinuous function \(f: \Bbb R \to\Bbb R\) such that for all real \(x\) and \(y\), \(f(x+y)=f(x)+f(y)\).

There is a Hamel basis for \(\Bbb R\) as a vector space over \(\Bbb Q\).

The set of all denumerable subsets of \(\Bbb R\) has power \(2^{\aleph_0}\).

If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\).

Weak Gelfand Extreme Point Theorem: If \(A\) is a non-trivial Gelfand algebra then the closed unit ball in the dual of \(A\) has an extreme point \(e\).  Morillon [1986].

There is an infinite, compact, Hausdorff, extremally disconnected topological space.  Morillon [1993].

Generalized Hahn-Banach Theorem:  Assume that \(X\) is a real vector space, \((Z,\preccurlyeq)\) is a Dedekind complete ordered vector space and \(X_0\) is a subspace of \(X\).  If \(\lambda_0 : X_0 \to Z\) is linear and \(p: X\to Z\) is sublinear and if \(\lambda_0 \preccurlyeq p\) on \(X_0\) then \(\lambda_0\) can be extended to a linear map \(\lambda : X\to Z\) such that \(\lambda \preccurlyeq p\) on \(X\). \ac{Schechter} \cite{1996b}

(For \(n\in\omega\), \(n\ge 2\).) \(PC(\aleph_0,n,\infty)\): Every denumerable set of \(n\)-element sets has an infinite subset with a choice function.

\(UT(\aleph_0,n,\aleph_0)\) for \(n\in\omega -\{0,1\}\): The union of a denumerable set of \(n\)-element sets is denumerable.

Tietze-Urysohn Extension Theorem: If \((X,T)\) is a normal topological space, \(A\) is closed in \(X\), and \(f: A\to [0,1]\) is continuous, then there exists a continuous function \(g: X\to [0,1]\) which extends \(f\).

Restricted Kinna Wagner Principle:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) and a function \(f\) such that for every \(z\subseteq Y\), if \(|z| \ge 2\) then \(f(z)\) is a non-empty proper subset of \(z\).

Restricted Ordering Principle:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that \(Y\) can be linearly ordered.

Restricted Choice for Families of Well Ordered Sets:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that the family of non-empty well orderable subsets of \(Y\) has a choice function.

\(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\).

\(PC(\infty,WO,\infty)\):  For every infinite family of non-empty well orderable sets, there is an infinite subfamily \(Y\) of \(X\) which has a choice function.

DUM:  The disjoint union of metrizable spaces is metrizable.

DUMN:  The disjoint union of metrizable spaces is normal.

Closed Filter Extendability for \(T_1\) Spaces: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter.

Countable Ultrafilter Theorem:  Every proper filter with a countable base over a set \(S\) (in \({\cal P}(S)\)) can be extended to an ultrafilter.

Every B compact (pseudo)metric space is Baire.

DPO:  Every infinite set has a non-trivial, dense partial order.  (A partial ordering \(<\) on a set \(X\) is dense if \((\forall x, y\in X)(x \lt y \to (\exists z \in X)(x \lt z \lt y))\) and is non-trivial if \((\exists x,y\in X)(x \lt y)\)).

Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.

\(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function.  \ac{Keremedis} \cite{1999b}.

Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. Ash [1983].

\(C(\infty,LO)\): Every set of non-empty linearly orderable sets has a choice function.

\(C(LO,LO)\): Every linearly ordered set of linearly orderable sets has a choice function.

\(C(LO,WO)\): Every linearly ordered set of non-empty well orderable sets has a choice function.

\(C(WO,LO)\): Every well ordered set of non-empty linearly orderable sets has a choice function.

\(MC(LO,LO)\): For each linearly ordered family of non-empty linearly orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).

\(MC(LO,WO)\): For each linearly ordered family of non-empty well orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).

\(MC(WO,LO)\): For each well ordered family of non-empty linearly orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).

\(KW(LO,\infty)\), The Kinna-Wagner Selection Principle for a linearly ordered family of sets: For every linearly ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

\(KW(\infty,LO)\), The Kinna-Wagner Selection Principle for a set of linearly orderable sets: For every set of linearly orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

\(KW(LO,LO)\), The Kinna-Wagner Selection Principle for a linearly ordered set of linearly orderable sets: For every linearly ordered set of linearly orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

\(KW(LO,<\aleph_0)\), The Kinna-Wagner Selection Principle for a linearly ordered set of finite sets: For every linearly ordered set of finite sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

\(KW(WO,LO)\), The Kinna-Wagner Selection Principle for a well ordered set of linearly orderable sets: For every well ordered set of linearly orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

\(KW(LO,WO)\), The Kinna-Wagner Selection Principle for a linearly ordered set of well orderable sets: For every linearly ordered set of well orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

Every infinite set can be partitioned into infinitely many sets, each of which has at least two elements. Ash [1983].

Every infinite set can be partitioned into sets each of which is countable and has at least two elements.

The product of compact Hausdorf spaces is countably compact. Alas [1994].

If \(\{f_i: i\in I\}\) is a family of functions such that for each \(i\in I\), \(f_i\subseteq E\times W\), where \(E\) and \(W\) are non-empty sets, and \(\cal B\) is a filter base on \(I\) such that

1. For all \(B\in\cal B\) and all finite \(F\subseteq E\) there is an \(i\in I\) such that \(f_i\) is defined on \(F\), and
2. For all \(B \in\cal B\) and all finite \(F\subseteq E\) there exist at most finitely many functions on \(F\) which are restrictions of the functions \(f_i\) with \(i\in I\),  

Suppose \((G,\Gamma)\) is a locally finite graph (i.e. \(G\) is a non-empty set and \(\Gamma\) is a function from \(G\) to \(\cal P(G)\) such that for each \(x\in G\), \(\Gamma(x)\) and \(\Gamma^{-1}\{x\}\) are finite), \(K\) is a finite set of integers, and \(T\) is a function mapping subsets of \(K\) into subsets of \(K\). If for each finite subgraph \((A,\Gamma_A)\) there is a function \(\psi\) such that for each \(x\in A\), \(\psi(x)\in T(\psi[\Gamma_A(x)])\), then there is a function \(\phi\) such that for all \(x\in G\), \(\phi(x)\in T(\phi[\Gamma(x)])\).

RC (Reflexive Compactness): The closed unit ball of a reflexive normed space is compact for the weak topology.

RCuc (Reflexive Compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is compact for the weak topology.

RCh (Reflexive Compactness for Hilbert spaces): The closed unit ball of a Hilbert space is compact for the weak topology.

Every infinite set \(S\) is the union of a set, well-ordered by inclusion, of subsets which are non-equipollent to \(S\).  

Every \(\cal W\)-frame is a \(\cal D\)-frame.  \ac{Ern\'e} \cite{2000}

Every \(\cal W\)-compactly generated complete lattice is algebraic.  \ac{Ern\'e} \cite{2000}.

Every non-compact topological space \(S\) is the union of a set that is well-ordered by inclusion and consists of open proper subsets of \(S\).

On every non-trivial Banach space there is a non-trivial linear functional (bounded or unbounded).

DUM(\(\aleph_0\)): The countable disjoint union of metrizable spaces is metrizable.

UT(\(\aleph_0\),cuf,cuf): The union of a denumerable set of cuf sets is cuf. (A set is cuf if it is a countable union of finite sets.)

\(UT(\aleph_0\),\(\aleph_0\),cuf): The union of a denumerable set of denumerable sets is cuf.

\(UT(WO,n,WO)\), \(n\in \omega-\{0,1\}\): The union of a well ordered set of \(n\) element sets can be well ordered.

\(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in  \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function.

If \((X,\cal T) \) is a first countable topological space and \((\cal B(x))_{x\in X}\) is a family such that for all \(x \in X\), \(\cal B(x)\) is a local base at \(x\), then there is a family \(( \cal V(x))_{x\in X}\) such that for every \(x \in X\), \(\cal V(x)\) is a countable local base at \(x\) and \(\cal V(x) \subseteq \cal B(x)\).

(Where \(p\) is a prime) \(AL21\)\((p)\): Every vector space over \(\mathbb Z_p\) has the property that for every subspace \(S\) of \(V\), there is a subspace \(S'\) of \(V\) such that \(S \cap S' = \{ 0 \}\) and \(S \cup S'\) generates \(V\) in other words such that \(V = S \oplus S'\).   Rubin, H./Rubin, J [1985], p.119, AL21.