A remark on $β$-locally closed sets

A remark on locally direct product subsets in a topological Cartesian space

Let $X$ and $Y$ be topological spaces. Let $C$ be a path-connected closed set of $X\times Y$. Suppose that $C$ is locally direct product, that is, for any $(a,b)\in X\times Y$, there exist an open set $U$ of $X$, an open set $V$ of $Y$, a subset $I$ of $U$ and a subset $J$ of $V$ such that $(a,b) \in U\times V$ and $$C\cap (U\times V)=I\times J$$ hold. Then, in this memo, we show that $C$ is globally so, that is, there exist a subset $A$ of $X$ and a subset $B$ of $Y$ such that $$C=A\times B$$ holds. The proof is elementary. Here, we note that one might be able to think of a (perhaps, open) si…

Locally closed sets and submaximal spaces

A topological space $X$ is called submaximal if every dense subset of $X$ is open. In this paper, we show that if $βX$, the Stone-Čech compactification of $X$, is a submaximal space, then $X$ is a compact space and hence $βX=X$. We also prove that if $\upsilon X$, the Hewitt realcompactification of $X$, is submaximal and first countable and $X$ is without isolated point, then $X$ is realcompact and hence $\upsilon X=X$. We observe that every submaximal Hausdorff space is pseudo-finite. It turns out that if $\upsilon X$ is a submaximal space, then $X$ is a pseudo-finite $μ$-compact space. An ex…

The weights of closed subgroups of a locally compact group

Let $G$ be an infinite locally compact group and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le w(G)$ for the weight $w(G)$ of $G$. It is shown that there is a closed subgroup $N$ of $G$ with $w(N)=\aleph$. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group $G$ and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le \lw(G)$, where $\lw(G)$ is the local weight of $G$, there are either no infinite compact subgroups at all or there is a compact subgroup $N$ of $G$ with $w(N)=\aleph$. (3) For an infinite abe…

Solid-set functions and topological measures on locally compact spaces

A topological measure on a locally compact space is a set function on open and closed subsets which is finitely additive on the collection of open and compact sets, inner regular on open sets, and outer regular on closed sets. Almost all works devoted to topological measures, corresponding non-linear functionals, and their applications deal with compact spaces. The present paper is one in a series that investigates topological measures and corresponding non-linear functionals on locally compact spaces. Here we examine solid and semi-solid sets on a locally compact space. We then give a method …

A Note on Locally Compact Subsemigroups of Compact Groups

An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every commutative cancellative pseudocompact locally compact Hausdorff topological semigroup with open shifts is a compact topological group.

On locally LC-spaces

A topological space $(X,τ)$ is called a locally LC-space if every point of $X$ has a neighborhood $U$ such that every Lindelöf subset of $(U,τ|U)$ is a closed subset of $(U,τ|U)$. The aim of this paper is to continue the study of locally LC-spaces.

一个非清醒的核心紧的交连续dcpo

A non-sober core-compact meet-continuous dcpo

本文构造了一个交连续dcpo，其Scott空间核心紧但非清醒、非局部紧，该dcpo是局部紧清醒完全格的真子dcpo，其Scott闭集格与原格同构，表明局部紧清醒dcpo不由Scott闭集格唯一确定。

通过线性细胞自动机对局部有限群的闭图像刻画

Closed Image Characterizations of Locally Finite Groups via Cellular Automata

证明群G是局部有限的当且仅当每个线性细胞自动机V^G→V^G在prodiscrete拓扑下具有闭图像，解决了Ceccherini-Silberstein和Coornaert的开放问题7。

When degree of roughness is a neighborhood over locally solid Riesz spaces

In this paper we introduce the notion of rough weighted $\mathcal{I}_τ$-limit points set and weighted $\mathcal{I}_τ$-cluster points set in a locally solid Riesz space which are more generalized version of rough weighted $\mathcal{I}$-limit points set and weighted $\mathcal{I}$-cluster points set in a $θ$-metric space respectively. Successively to compare with the following important results of Fridy [Proc. Amer. Math. Soc. {118} (4) (1993), 1187-1192] and Das [Topology Appl. {159} (10-11) (2012), 2621-2626], respectively be stated as \begin{description} \item[(i)] Any number sequence $x=\{x_{…

The Glueing Construction and Double Categories

We introduce Artin-Wraith glueing and locally closed inclusions in double categories. Examples include locales, toposes, topological spaces, categories, and posets. With appropriate assumptions, we show that locally closed inclusions are exponentiable, and the exponentials are constructed via Artin-Wraith glueing. Thus, we obtain a single theorem establishing the exponentiability of locally closed inclusions in these five cases.

可计算局部紧群及其闭子群

Computably locally compact groups and their closed subgroups

本文证明可计算局部紧群的闭子群 Chabauty 空间具有典范有效闭表示，并利用可数交广群刻画全不连通情形，最后证明包含特定闭子群的此类群的索引集是算术的。

局部壳拓扑

Locally hulled topologies

本文研究群拓扑由伪范数生成的问题，证明具有某种闭集基的拓扑可由球同样闭的伪范数族生成，并用多类拓扑例子说明其统一性。

Embedding into connected, locally connected topological gyrogroups

In this paper, it is proved that every topological gyrogroup $G$ is topologically groupoid isomorphic to a closed subgyrogroup of a connected, locally connected topological gyrogroup $G^{\bullet}$.

PFA(S)[S] and Locally Compact Normal Spaces

We examine locally compact normal spaces in models of form PFA(S)[S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of omega_1 and in which all separable closed subspaces are Lindelof.

Remarks on countable tightness

Countable tightness may be destroyed by countably closed forcing. We characterize the indestructibility of countable tightness under countably closed forcing by combinatorial statements similar to the ones Tall used to characterize indestructibility of the Lindelof property under countably closed forcing. We consider the behavior of countable tightness in generic extensions obtained by adding Cohen reals. We show that certain classes of well-studied topological spaces are indestructibly countably tight. Stronger versions of countable tightness, including selective versions of separability, are…

On Some Topological Concepts of TVS-Cone Metric Spaces and Fixed Point Theory Remarks

We prove that every TVS-cone metric space (i.e a cone metric space over a locally convex topological vector space $E$) is first countable paracompact topological space and by using Du's results in " [A note on cone metric fixed point theory and its equivalence, {Nonlinear Analysis},72(5),2259-2261 (2010)]", we conclude that every TVS-cone metric space is topologically isomorphic to a topological metric space. We also show how to construct comparable metric topologies to TVS-cone metric topologies by using the system of seminorms generating the topology of the locally convex topological vector …

Deficient topological measures on locally compact spaces

Topological measures and quasi-linear functionals generalize measures and linear functionals. We define and study deficient topological measures on locally compact spaces. A deficient topological measure on a locally compact space is a set function on open and closed subsets which is finitely additive on compact sets, inner regular on open sets, and outer regular on closed sets. Deficient topological measures generalize measures and topological measures. First we investigate positive, negative, and total variation of a signed set function that is only assumed to be finitely additive on compact…

Transitive actions of locally compact groups on locally contractible spaces

Suppose that $X=G/K$ is the quotient of a locally compact group by a closed subgroup. If $X$ is locally contractible and connected, we prove that $X$ is a manifold. If the $G$-action is faithful, then $G$ is a Lie group.