Induced Homeomorphism and Atsuji Hyperspaces

Ajit Kumar Gupta; Saikat Mukherjee

Home/Archive/arXiv

arXiv · 2205.11842

Induced Homeomorphism and Atsuji Hyperspaces

Ajit Kumar Gupta·Saikat Mukherjee

View original source ↗Read source document ↗Export metadata ↓

Abstract

Given uniformly homeomorphic metric spaces $X$ and $Y$, it is proved that the hyperspaces $C(X)$ and $C(Y)$ are uniformly homeomorphic, where $C(X)$ denotes the collection of all nonempty closed subsets of $X$, and is endowed with Hausdorff distance. Gerald Beer has proved that the hyperspace $C(X)$ is Atsuji when $X$ is either compact or uniformly discrete. An Atsuji space is a generalization of compact metric spaces as well as of uniformly discrete spaces. In this article, we investigate the space $C(X)$ when $X$ is Atsuji, and a class of Atsuji subspaces of $C(X)$ is obtained. Using the obtained results, some fixed point results for continuous maps on Atsuji spaces are obtained.

Explore related subjects

math.GN

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Ajit Kumar Gupta, Saikat Mukherjee. 2022-05-24. Induced Homeomorphism and Atsuji Hyperspaces. https://arxiv.org/abs/2205.11842

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The circle as a topological fractal

We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasová and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every $\varepsilon>0$ there are two continuous self-maps of the circle, depending on $\varepsilon$, whose images cover the circle and an integer $N$ such that every composition of $N$ of them has image of diameter less than $\varepsilon$. Thus two maps suffice at any prescribed scale, but no fixed pair works at all scales.

math.GN↗

arXiv2026

Finite-Point Metrizable Coarsenings: Compatible Gauges, Simplicial Metrics, and Hausdorff Lower Bounds

Let $(X,τ)$ be metrizable and let $F=\{a_1,\ldots,a_k\}\subseteq X$, where $2\le k<\infty$. We represent all metrizable topologies $σ\subseteqτ$ agreeing with $τ$ on $X\setminus F$ by compatible systems of continuous gauges $s_i:X\to[0,1]$ with $s_i^{-1}(0)=\{a_i\}$. The condition $\inf_X\max\{s_i,s_j\}>0$ for $i\ne j$ is equivalent to both Hausdorffness and metrizability of the prescribed gauge topology. A normalized product map into the standard simplex gives an explicit metric; its triangle inequality follows from a simplex slack inequality. This metric is complete whenever the auxiliary bounded compatible metric is complete. For two compatible systems, their coordinatewise minimum describes the intersection topology. It is compatible exactly when the two coarsenings have a common Hausdorff lower bound; in that case the intersection is metrizable and is their meet. Otherwise every common lower topology is non-Hausdorff. A closed-discrete construction produces such an obstructed pair for every noncompact metrizable space and every finite exceptional set with at least two points. Consequently, for these exceptional sets, the family is downward directed, or is a lattice, if and only if $(X,τ)$ is compact, in which case it consists only of $τ$.

math.GN↗

arXiv2026

Journey into special $T_1$-spaces

In this survey, we review certain types of special $T_{1}$ spaces and their associated fixed-point theorems. Kupka introduced the notion of a feeble topological contraction, which naturally generalizes Lipschitz contractions defined on metric spaces. Specifically, Kupka established a fixed-point theorem for feeble topological contractions possessing a closed graph within the product of arbitrary $T_{0}$ spaces. Furthermore, the $T_{1}$ separation axiom is shown to guaranty the uniqueness of such fixed points. Finally, we discuss peripheral Hausdorff and locally Hausdorff spaces within the context of these fixed-point results.

math.GN↗