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On subcompactness and countable subcompactness of metrizable spaces in ZF

Kyriakos Keremedis (2022)

Commentationes Mathematicae Universitatis Carolinae

We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space 𝐗 = ( X , T ) is countably compact if and only if it is countably subcompact relative to T . (iii) For every metrizable space 𝐗 = ( X , T ) , the following are equivalent: (a) 𝐗 is compact; (b) for every open filter ℱ of 𝐗 , ⋂ { F ¯ : F ∈ ℱ } ≠ ∅ ; (c) 𝐗 is subcompact relative to T . We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable...

On the additivity of the cardinalities of fuzzy sets of type II.

Ronald R. Yager (1983)

Stochastica

In this short note we show that for fuzzy sets of type II the additive rule for cardinalities holds true. The proof of this result requires an application of approximate reasoning as means of inference by use of the entailment principle.

On the bounding, splitting, and distributivity numbers

Alan S. Dow, Saharon Shelah (2023)

Commentationes Mathematicae Universitatis Carolinae

The cardinal invariants 𝔥 , 𝔟 , 𝔰 of 𝒫 ( ω ) are known to satisfy that ω 1 ≤ 𝔥 ≤ min { 𝔟 , 𝔰 } . We prove that all inequalities can be strict. We also introduce a new upper bound for 𝔥 and show that it can be less than 𝔰 . The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).

On the central limit theorem on IFS-events.

Jozefina Petrovicová, Riecan Beloslav (2005)

Mathware and Soft Computing

A probability theory on IFS-events has been constructed in [3], and axiomatically characterized in [4]. Here using a general system of axioms it is shown that any probability on IFS-events can be decomposed onto two probabilities on a Lukasiewicz tribe, hence some known results from [5], [6] can be used also for IFS-sets. As an application of the approach a variant of Central limit theorem is presented.

On the Compactness and Countable Compactness of 2 ℝ in ZF

Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " 2 ℝ is countably compact" and " 2 ℝ is compact"

On the complemented subspaces of the Schreier spaces

I. Gasparis, D. Leung (2000)

Studia Mathematica

It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space X ξ generated by subsequences ( e l n ξ ) and ( e m n ξ ) , respectively, of the natural Schauder basis ( e n ξ ) of X ξ are isomorphic if and only if ( e l n ξ ) and ( e m n ξ ) are equivalent. Further, X ξ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of ( e n ξ ) . It is also shown that there exists a complemented subspace spanned by a block basis of ( e n ξ ) , which is not isomorphic to a subspace generated by a subsequence of ( e n ζ ) , for every 0 ≤ ζ ≤ ξ ....

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