Dense embeddings in pathwise connected spaces
type: Item
instance of: scholarly article
title: Dense embeddings in pathwise connected spaces
zbMATH Open document ID: 0928.54017
DOI: 10.1016/S0166-8641(98)00016-9
author: Alessandro Fedeli
author: Attilio Le Donne
published in: Topology and its Applications
publication date: 2000-01-09T00:00:00Z
review text: A (pathwise) connectification of a space \(X\) is a (pathwise) connected topological space \(Y\) containing a dense (homeomorphic) copy of \(X\). \textit{S. Watson} and the reviewer [Houston J. Math. 19, No.~3, 469-481 (1993; Zbl 0837.54012))] have shown that the non-existence of isolated points is a necessary and sufficient condition for the existence of a \(T_1\) connectification of a \(T_1\)-space and a \(T_2\) connectification of a countable Hausdorff space. This paper studies the existence of pathwise connectifications and it is shown that the same condition (of being dense-in-itself) characterizes the existence of \(T_1\) pathwise connectifications of \(T_1\)-spaces and \(T_2\) pathwise connectifications of countable, first countable Hausdorff spaces. Examples are given of a countable Hausdorff space and a subspace of the real line having \(T_2\) connectifications, but which have no \(T_2\) pathwise connectification. A number of open problems are posed.
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54C25
zbMATH DE Number: 1340406
zbMATH Keywords: pathwise connectification
zbMATH Keywords: pathwise connected space
zbMATH Keywords: locally connected space
zbMATH Keywords: Euclidean space
reviewed by: Richard G. Wilson
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1340406