On removing one point from a compact space

脆弱空间

Flimsy Spaces

本文研究了$n$-脆弱空间的性质，证明对于$n≥3$不存在此类空间，并探讨了基于路径连通性的变种定义。

关于极不连通紧空间中若干特殊点的注记

Remarks on some special points in extremally disconnected compact spaces

在ZFC下证明极不连通紧空间中存在非可数离散子集极限点的特殊点。

$\mathcal I^K$-limit points, $\mathcal I^K$-cluster points and $\mathcal I^K$-Frechet compactness

In 2011, the theory of $\mathcal I^K$-convergence gets birth as an extension of the concept of $\mathcal{I}^*$-convergence of sequences of real numbers. $\mathcal I^K$-limit points and $\mathcal I^K$-cluster points of functions are introduced and studied to some extent, where $\mathcal{I}$ and $\mathcal{K}$ are ideals on a non-empty set $S$. In a first countable space set of $\mathcal I^K$-cluster points is coincide with the closure of all sets in the filter base $\mathcal{B}_f(\mathcal{I^K})$ for some function $f : S\to X$. Frechet compactness is studied in light of ideals $\mathcal{I}$ and $…

Compactification of cut-point spaces

We show that if $X$ is a separable locally compact Hausdorff connected space with fewer than $\mathfrak c$ non-cut points, then $X$ embeds into a dendrite $D\subseteq \mathbb R ^2$, and the set of non-cut points of $X$ is a nowhere dense $G_δ$-set. We then prove a Tychonoff cut-point space $X$ is weakly orderable if and only if $βX$ is an irreducible continuum. Finally, we show every separable metrizable cut-point space densely embeds into a reducible continuum with no cut points. By contrast, there is a Tychonoff cut-point space each of whose compactifications has the same cut point. The exam…

Connectifying a space by adding one point

It is a classical theorem of Alexandroff that a locally compact Hausdorff space has a one-point Hausdorff compactification if and only if it is non-compact. The one-point Hausdorff compactification is indeed obtained by adding the so called "point at infinity." Here we consider the analogous problem of existence of a one-point connectification, and keeping analogy, we prove that a locally connected normal space has a one-point normal connectification if and only if it has no compact component.

On compactness and fixed point theorems in partial metric spaces

In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that Hausdorff compact partial metric spaces are metrizable. In the second part of this article we discuss the significance of bottom sets of partial metric spaces in fixed point theorems for mappings acting in these spaces.