Statement:
\(MC(\infty,\infty)\) \((MC)\), 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).
Howard_Rubin_Number: 67
Parameter(s): This form does not depend on parameters
This form's transferability is: Not Transferable
This form's negation transferability is: Negation Transferable
Article Citations:
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 67 A | \(A(A2,H2)\): For every \(T_2\) topological space \((X,T)\)if \((X,T)\) is A2, then \((X,T)\) is hereditarily \(A2\). (\((X,T)\) is A2 meansif \(U\subseteq T\) covers \(X\) then \(\exists f:X\rightarrow T\) such thatfor all \(x\in X\), \(x\in f(x)\) and \(f\)"\(X\) refines \(U\).)Brunner [1983d] and Note 26. |
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| 67 B | \(MW\): Each set can be covered by a well orderedfamily of finite sets. Levy [1962]. |
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| 67 C | \(A(P)\): (Depends on \(P\in\{D1,B2,B3,H2,A2\}.)\)Every \(T_2\) topological space has property \(P\). Brunner [1983d]and Note 26. |
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| 67 D | For every Abelian group \(A\), every torsion freedivisible subgroup of \(A\) is a direct summand of \(A\). Armbrust [1972] and Note 24. |
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| 67 E | For any algebra {\bf A} with finitary operations(that is, {\bf A} is of type \(\tau\) where \(\tau = (K_i)_{i\in I}\) with\(K_i\) finite for each \(i\in I\)) and for any generating subset \(Z\) of{\bf A}, there is a function \(h: A\to \cal P_\omega(Z)\) (\(A\) is thedomain of {\bf A} and \(\cal P_\omega(Z)\) is the set of all finitesubsets of \(Z\).) such that \(\forall a\in A\), \(a\) is in thesubalgebra of {\bf A} generated by \(h(a)\). Diener [1989] andNote 115. |
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| 67 F | For any algebra {\bf A} with one unary operation \(f\)and any generating set \(Z\) of {\bf A}, there isa function \(h: A\to \cal P_\omega(Z)\) (\(A\) is the domain of {\bf A} and\(\cal P_\omega(Z)\) is the set of all finite subsets of \(Z\).) such that\(\forall a\in A\), \(a\) is in the subalgebra of {\bf A} generated by\(h(a)\). Diener [1989] and Note 115. |
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| 67 G | For every \(T_1\) topological space \((X,T)\) and everycellular family \(\cal O=\{O_i: i\in k\}\subseteq T - \{\emptyset\}\),there is a closed discrete set \(\cal I\) such that for every \(i\in k\),\(\cal I\cap O_i\ne\emptyset\). Keremedis [1998a] and Note 77. |
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| 67 H | Let \((X, T)\) be a \(T_4\) topological space and let\(\{G_i: i\in K\}\) be a locally finite family of pairwise disjoint,non-empty, open subsets of \(X\). Then there exists a continuous realvalued function \(f: X\to \Bbb R\) such that for each \(i\in K\), \(f\) vanisheson the boundary of \(G_i\) and \(f(x_i) = 1\) for some \(x_i\in G_i\).Keremedis [1997] and Note 43. |
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| 67 I | If \((X,T)\) is a \(T_4\) topological space and \(U =\{U_i: i\in K\}\) is a family of non-empty open sets, then there is afamily \(F = \{f_i: i\in K\}\) of non-negative, non-constant,continuous real valued functions on \(X\) such that each \(f_i\)vanishes on \(X - U_i\). Keremedis [1997]. |
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| 67 J | If \((X,T)\) is a \(T_4\) topological space and \(U =\{U_i: i\in K\}\) is a family of non-empty open sets, then there is afamily \(V = \{V_i: i\in K\}\) of non-empty closed (compact) sets suchthat \(U_i \supseteq V_i\) for each \(i\in K\). Keremedis [1997]. |
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| 67 K | DUN: The disjoint union of normal topologicalspaces is normal. Howard\slash Keremedis/Rubin/Rubin [1998a] andNote 141. |
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| 67 L | DUU: If \(C\) is a collection of pairwisedisjoint topological spaces each of which satisfies Urysohn'slemma, then the disjoint union of the spaces in \(C\) satisfiesUrysohn's lemma. Howard/Keremedis/Rubin/Rubin [1998a]and Note 141. |
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| 67 M | DUT: If \(C\) is a collection of pairwisedisjoint topological spaces each of which satisfies the Tietze-Urysohnextension theorem, then the disjoint union of the spaces in \(C\)satisfies the Tietze-Urysohn extension theorem.Howard\slash Kereme\-dis/Rubin/Rubin [1998a] and Note 141. |
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| 67 N | DUP: The disjoint union of paracompact spaces isparacompact. Howard\slash Keremedis/Rubin/Rubin [1998b] andNote 141. |
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| 67 O | DUMET: The disjoint union of metacompact spaces ismetacompact. Howard/Keremedis/Rubin/Rubin [1998b] and Note 141. |
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| 67 P | DUPFCS: The disjoint union of PFCS spaces is PFCS.(\(X\) is a PFCS space iff every point finite open cover of \(X\) isshrinkable.) Howard/Keremedis\slash Rubin/Rubin [1998b] andNote 141. |
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| 67 Q | DUPN: The disjoint union of paracompact spaces isnormal. Howard\slash Keremedis/Rubin/Rubin [1998b] and Note 141. |
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| 67 R | DUcwH: The disjoint union of cwH spaces is cwH.Howard/Keremedis\slash Rubin/Rubin [1998b] and Note 141. |
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| 67 S | DUcwN: The disjoint union of cwN spaces is cwN.Howard/Keremedis\slash Rubin/Rubin [1998b] and Note 141. |
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| 67 T | DUcwNN: The disjoint union of cwN spaces is normal.Howard\slash Keremedis/Rubin/Rubin [1998b] and Note 141. |
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| 67 U | Every open cover \(\cal U\) of a metric space \((X,d)\) canbe written as a well ordered union \(\bigcup \{U_\alpha : \alpha\in\gamma\}\)where \(\gamma\) is an ordinal and each \(U_\alpha\) is locally finite.Howard/Keremedis/Rubin/Stanley [1999] and Note 141. |
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| 67 V | Every open cover \(\cal U\) of a metric space \((X,d)\) canbe written as a well ordered union \(\bigcup \{U_\alpha : \alpha\in\gamma\}\)where \(\gamma\) is an ordinal and each \(U_\alpha\) is point finite.Howard/Keremedis/slash Rubin/ slash Stanley [1999] and Note 141. |
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| 67 W | Every closed (open) filter in a \(T_1\) topologicalspace has a well orderable filter base. Keremedis/Tachtsis [1999b] and Note 10. |
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| 67 X | Every open ultrafilter in \(T_1\) topological spacehas a well orderable filter base. Keremedis/Tachtsis [1999b]and Note 10. |
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| 67 Y | For every set \(A\ne\emptyset\), every filtercontained in \(\cal P(A)\) has a well orderable filter base.Keremedis/Tachtsis [2000] and Note 10. |
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| 67 Z | Every closed (open) filter in a dense in itself\(T_1\) topological space has a well orderable filter base.Keremedis/Tachtsis [2000] and Note 10. |
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| 67 AA | If \((X,T)\) is a \(T_2\) topological space and\(\cal B\) is a lattice of closed sets, then every maximal \(\cal B\)filter has a well orderable filter base. Keremedis/Tachtsis [2000] and Note 10. |
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| 67 AB | Every complete lattice is constructively\(\cal F\)-complete. Ern\'e [2000] and Note 154 |
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| 67 AC | Every complete lattice is constructively \(\calU\)-complete. Ern\'e [2000] and Note 154 |
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| 67 AD | \(\exists F\) AL21\((F)\): There is a field \(F\) such thatevery vector space over \(F\) 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 \(V = S \oplus S'\).Bleicher [1964], Rubin, H.\/Rubin, J. [1985, pp.122,123, theorems 6.35 and 6.36]. |
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| 67 AE | \(\exists F\) of characteristic \(0\) such that AL21\((F)\):There is a field \(F\) of characteristic \(0\) such that every vector spaceover \(F\) has the property that for every subspace \(S\) of \(V\), there is asubspace \(S'\) of \(V\) such that \(S \cap S' = \{ 0 \}\) and \(S \cup S'\)generates \(V\) in other words such that \(V = S \oplus S'\). Bleicher [1964], Rubin, H.\/Rubin, J. [1985, pp. 122,123, theorems6.35 and 6.36]. |
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| 67 AF | AL21\((\mathbb Q)\): Every vector space over\(\mathbb Q\) has the property that for every subspace \(S\) of \(V\), there isa subspace \(S'\) of \(V\) such that \(S \cap S' = \{ 0 \}\) and \(S \cup S'\)generates \(V\) in other words such that \(V = S \oplus S'\). Bleicher [1964], Rubin, H.\/Rubin, J. [1985, pp. 122,123, theorems6.35 and 6.36].\medskip |
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