arXiv subjects

Eliza Wajch

Publications and source records attributed to Eliza Wajch.

21 records · Page 2Linked to original sources

arXiv2018

Hausdorff compactifications in ZF

For a compactification $αX$ of a Tychonoff space $X$, the algebra of all functions $f\in C(X)$ that are continuously extendable over $% αX$ is denoted by $C_α(X)$. It is shown that, in a model of $\textbf{ZF}$, it may happen that a discrete space $X$ can have non-equivalent Hausdorff compactifications $αX$ and $γX$ such that $% C_α(X)=C_γ(X)$. Amorphous sets are applied to a proof that Glicksberg's theorem that $βX\times βY$ is the Cech-Stone compactification of $X\times Y$ when $X\times Y$ is a Tychonoff pseudocompact space is false in some models of $\mathbf{ZF}$. It is noticed that if all Tychonoff compactifications of locally compact spaces had $C^{\ast}$-embedded remainders, then van Douwen's choice principle would be satisfied. Necessary and sufficient conditions for a set of continuous bounded real functions on a Tychonoff space $X$ to generate a compactification of $X$ are given in $\mathbf{ZF}$. A concept of a functional Čech-Stone compactification is investigated in the absence of the axiom of choice.

math.GN↗

arXiv2017

The minimizing vector theorem in symmetrized max-plus algebra

Assuming ZF and its consistency, we study some topological and geometrical properties of the symmetrized max-plus algebra in the absence of the axiom of choice in order to discuss the minimizing vector theorem for finite products of copies of the symmetrized max-plus algebra. Several relevant statements that follow from the axiom of countable choice restricted to sequences of subsets of the real line are shown. Among them, it is proved that if all simultaneously complete and connected subspaces of the plane are closed, then the real line is sequential. A brief discussion about semidenrites is included. Older known proofs in ZFC of several basic facts relevant to proximinal and Chebyshev sets in metric spaces are replaced by new proofs in ZF. It is proved that a nonempty subset C of the symmetrized max-plus algebra is Chebyshev in this algebra if and only if C is simultaneously closed and connected. An application of it to a version of the minimizing vector theorem for finite products of the symmetrized max-plus algebra is shown. Open problems concerning some statements independent of ZF and other statements relevant to Chebyshev sets are posed.

math.GN↗

arXiv2014

Compactness and compactifications in generalized topology

A generalized topology in a set $X$ is a collection $\text{Cov}_X$ of families of subsets of $X$ such that the triple $(X,\bigcup \text{Cov}_X,\text{Cov}_X)$ is a generalized topological space in the sense of Delfs and Knebusch. In this work, notions of topological and admissible compactness of generalized topologies are introduced to begin and investigate a theory of compactifications, in particular, of Wallman type in the category of weakly normal generalized topological spaces. Among other facts, we prove in ZF that the ultrafilter theorem (in abbreviation UFT) holds if and only if all Wallman extensions of every weakly normal generalized topological space are compact. In consequence, we develop the theory of compactifications in ZF+UFT when it is not necessary to use AC, while ZF is not enough.

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