In every vector space, every generating set contains a basis.

Every commutative ring with unit has a maximal ideal.

For all infinite cardinals \(m\), \(UT(m,m,m)\):If every member of an infinite set \(A\) has power \(|A|\) then\(|\bigcup A| = |A|\). (Note that [1 C] implies \(m^{2} =m\) forall infinite cardinals m.)

Cardinal successors 3: For every cardinal \(m\) there isa cardinal \(n\) such that \(m < n\) and \((\forall p)( m < p \rightarrow n \le  p )\).

\(K(0)\):  For every set \(S\) there is an ordinal \(\alpha\) and a one to one function \(f: S \rightarrow \alpha\). (See Form 81(\(n\))).

For all ordinals \(\alpha\), \(DC(\aleph_{\alpha})\): 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 \(\forall\beta < \aleph_{\alpha}\), \(\{f(\gamma):\gamma < \beta\}\mathrel Rf(\beta)\).

Form 163 + Form 8.

Form 164 + Form 67.

In every metric space \((X,d)\) there is an\(\epsilon\)-lattice for some \(\epsilon >0\) which is not a strict upperbound of \(d\).

Form 49 \(+\)Form 51.

Form 133 \(+\)Form 64.  Clear

Hajnal's Free Set Principle:  For every set mapping \(f: X\rightarrow [X]^{<\lambda}\) where \(\lambda\) is a well orderable cardinal and \(\lambda< |X|\) there is a free set of cardinality \(|X|\). ((\([X]^{<\lambda}\) is the set of subsets of \(X\) of cardinality \(<\lambda\)).

The ramification Lemma: If \(S: X^{\le\theta}\rightarrow {\cal P}(E)\) is a ramification system, then for each \(g \in E\) there is an \(f\) which is maximal (with respect to inclusion) in \(\{h \in X^{\le\theta}: g\in S(h)\}\).

For every set mapping \(f: X\rightarrow [X]^{WO}\)(well ordered subsets of \(X\)) there is a co-well-orderable free subset\(H\). (\(H\) is co-well-orderable if \(X\setminus H\) is well orderable).

\(A(D1,C)\): For every \(T_2\) topological space \((X,T)\),if \(X\) is covered by a well ordered family of closed, discrete sets theneach family \(U\subseteq T\) of pairwise disjoint sets is  well ordered.

\(A(P)\): (Depends on \(P\) for \(P\in\{W1,W2,B1,L1,L2,S,C,H1,A1\}\)) Every \(T_2\) topological space has property \(P\).

For any partial order \((X,\le)\) there is a \(\sup\) function on the bounded (above), well ordered subsets of \(X\).

For any partial order \((X,\le)\) there is a \(\sup\) function on the bounded subsets of \(X\).

For any partial order \((X,\le)\) if every well ordered subset of \(X\) is bounded then there is a \(\sup\) function on the well ordered subsets of \(X\).

For every directed partial order \((X,\le)\) there is a\(\sup\) function on the bounded, well ordered subsets of \(X\).

For every directed partial order, if every well ordered subset of \(X\) is bounded above then there is a \(\sup\) function on the well ordered subsets of \(X\).

For  every  partially  ordered  set \((X,\le )\),  if every well ordered subset is bounded above then every function \(f : X \rightarrow  X\) satisfying \(\forall t\in X\), \(t\le f(t)\) has a fixed point.

Form 133 + Form 88.

Form 15 + Form 91.

Refined Dual Cantor-Bernstein Theorem:  If \(f : X\rightarrow Y\) and \(g : Y \rightarrow  X\) are onto then there is a bijection \(h : X \rightarrow  Y\) such that \(h \subseteq f \cup g^{-1}\).

If \((P,\le)\) is a quasi-order and \(P \neq \emptyset\) and every antichain is finite, then there is a \(P\) generic filter which is well orderable.

Assume \(f: S\rightarrow R\) where \(f\) is surjective and \(R\subseteq S\).Then every mapping \(g\) of \(S\) into \(R\) has the form \(g = f\circ h\) for some \(h: S \rightarrow S\).

Assume \(f: S \rightarrow R\) where \(f\) is surjective and \(R\subseteq S\).Then the mappings of \(S\) onto \(R\) are precisely the mappings of the form \(f\circ h\)  where \(h: S\rightarrow S\) and \((\forall y\in R)(\exists x\in S)(h(x)\in f^{-1}(\{y\}))\).

Assume \(f: S\rightarrow R\) where \(R\subseteq S\). Then the retractions of \(S\) onto \(R\) (that is, the mappings \(g: S \rightarrow R\) which are the identity on \(R\)) are precisely the mappings \(f\circ h\) where \(h: S \rightarrow S\) and \((\forall y\in R)(h(y)\in f^{-1}(\{y\}))\).

For every class \(K\) of similar algebras \(SP^{g} K\supseteq P^{g}S\ K\).  (For any class \(K\) of similar algebras, \(S\ K\) is the class of all subalgebras of elements of \(K\) (not closed under isomorphism) and \(P^{g}K\) is the class of direct products of elements of \(K\) not closed under isomorphism).

There is a set \(S\) such that \(SVC\) holds with \(S\) and such that \({\cal P}(S) - \{\emptyset\}\) has a choice function.  Also see Form 191.

Form 191 + Form 8 + Form 193. Note 60.

Form 192  + Form 191.

\(E(V,VI)\): For all infinite cardinals \(m\), \(2m = m\) implies \(m^{2} = m\).

  \(E(VI,VII)\):  For all infinite cardinals \(m\), \(m^{2} = m\) implies there is an ordinal \(\alpha\) such that \(m =\aleph _{\alpha }\).

Let \({\cal F}\) be a family of mappings of a  poset \((X,\le )\) into itself such that \(\forall f\in{\cal F}\), \(\forall x\in X\),\(f(x)\le x\). If for some element \(e\) of \(X\) each chain in \(X\) containing \(e\) has a lower bound, then the family \({\cal F}\) has a common fixed point.

There is an ordinal \(\beta\) such that for all cardinals \(x\), \(\alpha(x+\aleph(x), x\cdot\aleph(x))<\beta\).

There is an ordinal \(\beta\) such that for all cardinals\(x\), \(\alpha(x,x^2) < \beta\).

There is an ordinal \(\beta\) such that for no cardinal \(x\) is there a strictly increasing sequence of cardinals \(\left \{ \,y_{\gamma} : \gamma < \beta \, \right \}\) such that \(y_{\gamma}^2 = x\).

Every lattice \(\langle L,\cap,\cup, \subseteq\rangle\)of sets (that is, \(L\) is a family of sets and \(\cap,\ \cup\) and \(\subseteq\) are the usual operations) with a greatest element has a maximal, proper ideal.

Any continuous surjection between compact spaces, where the range space is Hausdorff, has an irreducible restriction to a closed subset of its domain.

For any type \(\tau\) of the form \(\tau = (M)\), everyhomomorphism of an algebra of type \(\tau\) is closed.

For any type \(\tau\) of the form \(\tau = (M)\), every homomorphism of an algebra A of type \(\tau\) into an algebra B of type \(\tau\) maps A onto a subalgebra of B.

Let \(S\) be any family of disjoint non-empty sets andlet \(\tau = (\bigcup S)\), then any homomorphism from \(\frak P(\tau;\bigcup S)\)into \(\frak P(\tau;S)\) maps \(\frak P(\tau;\bigcup S)\) onto a subalgebra of \(\frak P(\tau;S)\).

Turinici's Fixed Point Theorem:  If \((X,\le)\) is a directed, partially ordered set and \(\tau\) is a topology on \(X\) such that

1. \((X,\le)\) is upper semicontinuous with respect to \(\tau\),
2. Every well ordered subset of \((X,\le)\) is convergent as a net,

For all cardinal numbers \(\kappa_1\) and \(\kappa_2\), if a graph is both \(\kappa_1\) and \(\kappa_2\) colorable then it is also\(\kappa\) colorable for some \(\kappa\) such that \(\kappa\le^*\kappa_1\) and \(\kappa\le^*\kappa_2\). (\(\le^*\) is the surjective cardinal ordering).

Every graph has an irreducible, good coloring.

Every graph of the form \(G(A,H(A))\) has an irreducible, good coloring.  (\(H\) is the Hartogs aleph function.)

Every graph has a chromatic number.

Birkhoff Representation Theorem: Every algebra isisomorphic to a subdirect product of subdirectly irreducible algebras.

For each \(\lambda\in\Lambda\) let \(S_\lambda\) be a subset of some topological space \(X_\lambda\). Then

For each \(\lambda\in\Lambda\), let \(X_\lambda\) be a complete uniform space. Then, the product \(\prod_{\lambda\in\Lambda}X_\lambda\) equipped with the product uniform structure, is also complete.(See, for example, Kelley [1955] pp 176ff for definitions.)

(Depends on the prime \(p\)) Every group has a maximal \(p\)-subgroup.

Strong Nielsen-Schreier Theorem II: If \(G\) is a group which is freely generated by \(X\) and \(U\) is a subgroup of \(G\),then there is a subset \(A\) of \(G\) which freely generates \(U\) and is level with respect to \(X\).

Assume \(P\) is a partially ordered set in which every non-empty chain has an upper bound and \(f: P\to P\) satisfies \(\forall x\in f(P) \cup \{\,a\in P : a\) is an upper bound for some chain in \(f(P)\,\}\),\(x\le f(f(x))\).  Then \(f\) has a fixed apex \(u\) (that is, there is some \(v\in P\) such that \(f(u)=v\) and \(f(v)=u\).)

Let \(P\) be a partially ordered set in which every increasing sequence has an upper bound and \(f:P\to P\) satisfies \(x \le f(f(x))\) for all \(x\in f(P) \cup \{x\in P : x\) is an upper bound for some increasing sequence in \(f(P)\}\). Then \(f\) has a fixed apex \(u\)(that is, \(\exists v\in P\), such that \(f(u) = v\) and \(f(v) = u\)). (For purposes of [1 BD], ``sequence'' means a function whose domain is an ordinal.)

Every unique factorization domain can be mapped homomorphically onto a non-zero subdirectly irreducible ring. Gratzer [1979] pp 122-124.

(\(DT_0)\): Every topological space \(X\) has a \(T_0\) subspace that is dense in \(X\). (\(Y\) is dense in \(X\) if there is no non-empty open subset \(O\subseteq X\) such that \(O\cap Y=\emptyset\).)

(C\(T_0\)): Every topological space \(X\) has a \(T_0\)subspace that is codense in \(X\). (\(Y\) is codense in \(X\) if there is nonon-empty closed subset \(C\subseteq X\) such that \(C\cap Y=\emptyset\).)

\((TT_0)\): Every topological space \(X\) has a \(T_0\) subspace that is thick in \(X\). (\(Y\) is thick in \(X\) if there is nonon-empty open and closed (clopen) subset \(H\subseteq X\) such that\(H\cap Y=\emptyset\).)

In every vector space over the two element field, every generating set contains a basis.

Every open cover \(\cal{U}\) of a metric space \((X,d)\) can be written as a well ordered union \(\bigcup\{U_\alpha :\alpha\in\gamma\}\) where \(\gamma\) is an ordinal and each \(U_\alpha\) is pairwise disjoint. Howard/Keremedis/Rubin/Stanley [1999].

Every family of closed sets in a \(T_2\) topological space includes a maximal subfamily with the finite intersectionproperty.  H.Rubin/J.Rubin [1985], p 60, \(M14\).

For every family \(\{(X_i,T_i): i\in k\}\) of \(T_1\) topological spaces, the box topology contains an element \(\prod_{i\in k}O_i\), such that \(\emptyset\ne O_i\) and \(O_i\ne X_i\) for all \(i\in k\).

For every family \(\{(X_i,T_i): i\in k\}\) of disjoint \(T_1\) topological spaces, the free union contains an open set \(O\) such that \(\emptyset\ne O\cap X_i\) and \( O\cap X_i\ne X_i\) for all\(i\in k\).

For every \(T_1\) topological space \((X,T)\) and every base (subbase) \(B\) for \(X\) there exists a \(\subseteq\)-maximal cellular family \(Q\subseteq B\).

For every dense in itself topological space \((X, T)\) and every base \(B\) for \(X\) there exists a nest \(N\) included in \(B\).

For every dense in itself topological space \((X,T)\) and every base \(B\) for \(X\) there exists a tower \(R\) included in \(B\).

For every \(T_2\) space \((X,T)\) and every subbase \(\cal B\) for \(X\), if \(\cal C\) and \(\cal D\) are cellular families of \(X\) included in \(\cal B\), then there is a cellular family \(\cal E\subseteq\cal B\) such that \(\cal C\) and \(\cal D\) are equipollent to subsets of\(\cal E\).

For every Abelian group \((G,+)\), for every \(H \subseteq G\), the infinite system of equations \[x_i + y_i = a_i,\ i\in I,\ a_i\in G\] has a solution in \(H\) iff each of its equations has a solution in \(H\).

For every Boolean ring \((B,+,\cdot)\) and every \(H \subseteq  B\)  which is closed under \(+\) there exists a \(\subseteq\)-maximal ideal \(Q\subseteq B\) such that \(H\cap Q=\{0\}\).

In every vector space over \({\Bbb Q}\), every generating  set includes a basis.

If \((X,T)\) is a \(T_4\) topological space and \(G =\{G_i: i\in K\}\), \(|G_i|\ge 2\), is a locally finite family of compact sets with non-empty pairwise disjoint interiors, then there is a continuous real valued function \(f\) on \(X\) which assumes at least two distinct values on each \(G_i\).

If \((X,T)\) is a \(T_4\) topological space and \(U =\{U_i: i\in K\}\), \(|U_i|\ge 2\), is a locally finite family ofopen non-empty pairwise disjoint sets, then there is a continuous real valued function \(f\) on \(X\) which assumes at least two distinct values on each \(U_i\).

For every Abelian group \(G\) and every subgroup \(H\) of \(G\) there exists a set of representatives for the quotient \(G/H\).

If \(x\) is a set of sets of cardinality at least 2, then there exists a function \(f\) such that for each \(u\in x\), \(f(u)\) is a finite, non-empty, proper subset of \(u\). (Form [1 BY] implies [62 E] (\(KW(\infty,<\aleph_0)\)) and Form 67 (MC) and Form 62 \(+\) Form 67 \(\to\) AC.)

Vector Space Multiple Choice:  For every family \(V = \{\,V_i : i \in K\,\}\) of non-trivial vector spaces there is a family \(F = \{\, F_i : i\in K\,\}\) such that for each \(i\in K\),\(F_i\) is a non-empty, finite, independent subset of \(V_i\).

Strong Nielsen-Schreier Theorem I:  If \(G\) is a groupwhich is freely generated by \(X\) and \(U\) is a subgroup of \(G\), then there is a subset \(A\) of \(G\) which freely generates \(U\) and has the Nielsen property with respect to \(X\).

For every infinite cardinal \(\kappa\), if \((X,T)\) is a \(T_2\) space having a base \(B\) such that no \(Q\subseteq B\), \(|Q| <\kappa\), is a tower, then for every family \(\cal D=\{D_i:i\in\kappa\}\) of dense open sets of \(X\) there is a filter \(F\subseteq T\) such thatfor every \(i\in\kappa\) there  is a \(b\in F\) with \(b\subseteq D_i\).

\(PC(\aleph_0,\hbox{odd},\infty)\) +\(MC_{\omega^+}\). \(PC(\aleph_0,\hbox{odd},\infty)\) is ``Every denumerable family of odd sized (finite) sets has an infinite subfamily with a choice function.'' and \(MC_{\omega^+}\)is ``For every natural number \(n \ge 1\) and forevery set of non-empty sets, there is a function \(f\) such that for each \(u\in x\), \(f(u)\) is a non-empty finite subset of \(u\)  suchthat \(|f(u)|\) and \(n\) are relatively prime.''

In a commutative ring, every proper ideal can be extended to a maximal ideal. ([1 CD] implies
[1 B]).

Every open cover \(\cal{U}\) of a metric space \((X,d)\)can be written as a well ordered union \(\bigcup\{U_\alpha :\alpha\in\gamma\}\) where \(\gamma\) is an ordinal and each \(U_\alpha\) is discrete. Howard/Keremedis/Rubin/Stanley [1999] .

If a topological space is \(cwH\), then it is \(cwH(B)\)for every basis \(B\).

The disjoint union of \(cwH\) spaces is \(cwH(B)\) for every basis \(B\).

If a topological space is \(cwN\), then it is \(cwH(B)\) for any basis \(B\).

Every compact space is A-U compact.

Every B compact Hausdorff space is A-U compact.

Vector Space Kinna-Wagner Principle: For every family \(V = \{V_i : i \in K\}\) of non-trivial vector spaces there is a family \(F = \{F_i : i\in K\}\) such that for each \(i\in K\), \(F_i\) is a non-empty, independent subset of \(V_i\).

Every complete lattice is constructively complete. \ac{Ern\'e} \cite{2000}

(Where \(\cal Z\) is a subset selection such that for all partial orders \((P,\le)\), \(\cal E P\subseteq \cal Z P\subseteq \cal D P\).) Every complete lattice is \(\cal Z^\lor\)-constructively complete. \ac{Ern\'e} \cite{2000}.

Every directed partially ordered set is constructively directed. \ac{Ern\'e} \cite{2000}.

For all \(X\), if \(X\neq\emptyset\), then there is a binary operation on \(X\) that makes \(X\) a group. Rubin, H./Rubin, J. [1985], p.110, AL14}.