Searcharxiv⌕ Search

arXiv subjects

Artur Piękosz

Publications and source records attributed to Artur Piękosz.

6 recordsLinked to original sources
arXiv2021

Stone duality for Kolmogorov locally small spaces

We prove three new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of compact (not necessarily Hausdorff) open sets as objects and spectral mappings respecting the decent lumps and satisfying a boundedness condition as morphisms as well as it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. Some theory of strongly locally spectral spaces is developed.

math.GN↗
arXiv2019

Locally small spaces with an application

We develop the theory of locally small spaces in a new simple language and apply this simplification to re-build the theory of locally definable spaces over structures with topologies.

math.GN↗
arXiv2018

Bornological quasi-metrizability in generalized topology

A concept of quasi-metrizability with respect to a bornology of a generalized topological space in the sense of Delfs and Knebusch is introduced. Quasi-metrization theorems for generalized bornological universes are deduced. A uniform quasi-metrizability with respect to a bornology is studied. The class of locally small spaces is considered and a possibly larger class of weakly locally small spaces is defined. The proofs and numerous examples are given in \textbf{ZF}. An example of a weakly locally small space which is not locally small is constructed under \textbf{ZF+CC}. Several categories, relevant to generalized bornological universes, are defined and shown to be topological constructs.

math.GN↗
arXiv2018

Compact complement topologies and k-spaces

Let $(X,τ)$ be a Hausdorff space, where $X$ is an infinite set. The compact complement topology $τ^{\star}$ on $X$ is defined by: $τ^{\star}=\{\emptyset\} \cup \{X\setminus M, \text{where $M$ is compact in $(X,τ)$}\}$. In this paper, properties of the space $(X, τ^{\star})$ are studied in $\mathbf{ZF}$ and applied to a characterization of $k$-spaces, to the Sorgenfrey line, to some statements independent of $\mathbf{ZF}$, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Among other results, it is proved that the axiom of countable multiple choice (\textbf{CMC}) is equivalent with each of the following two sentences: (i) every Hausdorff first countable space is a $k$-space, (ii) every metrizable space is a $k$-space. A \textbf{ZF}-example of a countable metrizable space whose compact complement topology is not first countable is given.

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↗