Statement:
\(C(\aleph_{0},\infty)\):
Howard_Rubin_Number: 8
Parameter(s): This form does not depend on parameters
This form's transferability is: Transferable
This form's negation transferability is: Negation Transferable
Article Citations:
Sierpi'nski-1918: L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 8 A | \(PC(\infty,\infty,\infty)\): Every infinite family ofnon-empty sets has an infinite subfamily with a choice function.Brunner [1983b]. |
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| 8 B | \(PC(\aleph_{0},\infty,\infty)\): Every denumerablefamily of non-empty sets has an infinite subfamily with a choicefunction. Monro [1975]. |
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| 8 C | Every \(\sigma\)-compact space is Lindel\"of.Brunner [1982b] and Note 43. |
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| 8 D | Every \(\sigma \)-compact, locally compact \(T_{2}\)space is weakly Lindel\"of. Brunner [1982b] and Note 43. |
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| 8 E | In a metric space every sequentially continuous realvalued function is continuous. Brunner [1982c]. |
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| 8 F | Every pseudometric space with a countable base isseparable. Bentley/Herrlich [1998] and Note 10. |
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| 8 G | Let \(R\) be a relation and \(R^*\), the transitive closureof \(R\). (\(x\mathrel {R^*}y\) iff there exists a finite ascending \(R\)-chainconnecting \(x\) to \(y\).) If every ascending \(R\)-chain is finite, then everyascending \(R^*\)-chain is finite. Diener [1994]. |
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| 8 H | A disjoint union of a countable set of compact\(T_2\) spaces is Lindel\"of. Keremedis [1998b] and Note 132. |
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| 8 I | A disjoint union of a countable set of Lindel\"of\(T_2\) spaces is Lindel\"of. Keremedis [1998b] and Note 132. |
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| 8 J | Hemicompact \(T_2\) spaces are Lindel\"of.(A hemicompact space is a space \((X,T)\) that can be writtenas a disjoint union \(\bigcup\{K_n: n\in\omega\}\) such that(i) each \(K_n\) is a compact set in \(X\);(ii) \(K_N\subseteq K_{n+1}\) for all \(n\in\omega\); and(iii) for every compact set \(K\) in \(X\), there is an \(n\in\omega\)such that \(K\subseteq K_n\).) Keremedis [1998b] and Note 132. |
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| 8 K | Hemilindel\"of \(T_2\) spaces are Lindel\"of.(In [8 J], in the definition of hemicompact replace ``compact''by ``Lindel\"of'' to get the definition of hemilindel\"of.)Keremedis [1998b] and Note 132. |
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| 8 L | Every second countable topological space is separable.Keremedis [1998b] and Note 132. |
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| 8 M | Form 131 +Form 32. Keremedis [1998b] and note132. |
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| 8 N | \(DP\): If \((X,T)\) is a topological space having acountable \(\pi\)-base, then for every family \(D=\{D_i: i\in\omega\}\) ofdense open sets of \(X\), there is a countable dense set \(S\subseteq X\) suchthat for all \(i\in\omega\) and for all but finitely many \(s\in S\),\(s\in D_i\). Keremedis [1998b] and Note 132. |
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| 8 O | \(DUCCC\) +Form 131.\(DUCCC\) is ``If \(X\) is a denumerable family of disjoint topologicalspaces each having the ccc property, then the disjoint union of thespaces in \(X\) has the ccc property.'' (\(Y\) has the ccc property ifevery collection of disjoint open sets is countable.)Keremedis [1998b] and Note 132. |
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| 8 P | \(DUSS\) +Form 131.\(DUSS\) is ``The disjoint union of a denumerable family of separabletopological spaces is separable.'' Keremedis [1998b] and note132. |
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| 8 Q | \(DUSC\) +Form 131.\(DUSC\) is ``The disjoint union of a denumerable family of secondcountable topological spaces is second countable.'' Keremedis [1998b] and Note 132. |
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| 8 R | \(SCDUL\) +Form 131.\(SCDUL\) is ``The denumerable disjoint union of second countablespaces is Lindel\"of.'' Keremedis [1998b] and Note 132. |
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| 8 S | \(PC(WO,\infty,\infty)\): Every infinite well orderedset of non-empty sets has an infinite subset with a choice function.([8 A] \(\to\) [8 S] \(\to\) [8 B]) |
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| 8 T | \(PUT(\aleph_0,\infty,\aleph_0)\): The union of adenumerable set of pairwise disjoint non-empty sets has a denumerablesubset. ([8 A] \(\to\) [8 T] \(\to\) [8 B]) |
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| 8 U | \(PUT(WO,\infty,WO)\): The union of an infinite wellordered set of pairwise disjoint non-empty sets has an infinite wellordered subset. ([8 U] \(\leftrightarrow\) [8 T]) |
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| 8 V | Every pseudometric Lindel\"of space is separable.Bentley/Herrlich [1998] and Note 10. |
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| 8 W | Subspaces of separable pseudometric spaces are separable.Bentley/Herrlich [1998] and Note 10. |
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| 8 X | Every sequentially bounded pseudometric space istotally bounded. Bentley/Herrlich [1998] and Note 10. |
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| 8 Y | Every totally bounded pseudometric space isseparable. Bentley/Herrlich [1998] and Note 10. |
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| 8 Z | Every sequentially bounded pseudometric space isseparable. Bentley/Herrlich [1998] and Note 10. |
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| 8 AA | Baire Category Theorem for Complete PseudometricSpaces with a countable base. Every non-empty complete pseudometricspace with a countable base is of the second category (non-meager).Bentley/Herrlich [1998] and Note 28. |
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| 8 AB | Baire Category Theorem for Complete TotallyBounded Pseudometric Spaces. Every non-empty complete totally boundedpseudometric space is of the second category (non-meager).Bentley/Herrlich [1998], and notes 6 and 28. |
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| 8 AC | Every sequentially compact pseudometric space istotally bounded. Bentley/Herrlich [1998] and notes 6 and 10. |
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| 8 AD | Every totally bounded, complete pseudometric spaceis compact. Bentley/Herrlich [1998] and notes 6 and 28. |
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| 8 AE | Every sequentially compact pseudometric space iscompact. Bentley/Herrlich [1998] and Note 10. |
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| 8 AF | In a pseudometric space, every infinite subset hasan accumulation point if and only if the space is complete and totallybounded. Bentley/Herrlich [1998] and Note 10. |
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| 8 AG | Compact pseudometric spaces are separable.Bentley/Herrlich [1998] and Note 10. |
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| 8 AH | In a metric space \(X\), if \(x\) is an accumulationpoint of a subset \(A\subseteq X\) then there is a sequence of elementsof \(A\) which converges to \(x\). Herrlich/Steprans [1997]. |
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| 8 AI | A function from one metric space to another iscontinuous if and only it is sequentially continuous.Herrlich/Steprans [1997]. |
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| 8 AJ | Countable products of pseudometric spaces areBaire. Herrlich/Keremedis [1999a], notes 10 and 28. |
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| 8 AK | Countable products of compact pseudometric spaces arecompact. Herrlich/Keremedis [1999a] and Note 10. |
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| 8 AL | For pseudometric spaces, every Cauchy filterconverges if and only if every Cauchy sequence converges.Herrlich/Keremedis [1999a] and Note 10. |
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| 8 AM | Every sequentially compact pseudometric space isBaire. Herrlich/Keremedis [1999], notes 6, 10 and 28. |
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| 8 AN | Every sequentially compact, totally boundedpseudometric space is Baire. Herrlich/Keremedis [1999], notes 6, 10 and 28. |
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| 8 AO | \(PUT(\aleph_0,\infty,WO)\): The union of adenumerable set of pairwise disjoint non-empty sets has an infinite wellordered subset. (8 \(\to\) [8 AO] \(\to\) [8 T]) |
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| 8 AP | Every super second countable topological space isseparable. Gutierres [2004] and Note 159. |
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| 8 AQ | If a topological space has a countable local base at apoint \(x\) then every local base at \(x\) contains a countable base at \(x\).Gutierres [2004] and Note 159. |
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| 8 AR | In a first countable topological space every localbase at a point \(x\) contains a countable base at \(x\). Gutierres [2004] and Note 159. |
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| 8 AS | A first countable topological space is Hausdorff ifand only if every sequence has at most one limit. Gutierres [2004]. |
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