Rectifiable spaces
type: Item
instance of: scholarly article
title: Rectifiable spaces
zbMATH Open document ID: 0844.22009
DOI: 10.1016/0166-8641(95)00027-5
author: Alexandra S. Gul'ko
published in: Topology and its Applications
publication date: 1996-03-20T00:00:00Z
review text: The author studies those spaces \(X\) which are rectifiable, i.e., for which there are a surjective homeomorphism \(\psi : X^2 \twoheadrightarrow X^2\) and a distinguished element \(e \in X\) such that every \(x \in X\) satisfies (a) \(\Psi (x,x) = (x,e)\) and (b) \(\Psi [\{x\} \times X] = \{x\} \times X\). Every topological group is rectifiable and every rectifiable space is homogeneous; both converse statements fail. The thrust of the paper is that several familiar assertions about topological groups remain true for rectifiable spaces. For example, first countability yields metrizability. The author shows further, with \(w, \chi, \pi w, \pi \chi\), \(d\) and \(k\) denoting respectively weight, character, \(\pi\)-weight, \(\pi\)-character, density character and compact-covering number, that the following relations are valid for arbitrary rectifiable spaces \(X : \pi \chi (X) = \chi (X)\); \(w(X) = d(X) \cdot \chi (X)\); \(w(X) \leq k(X) \cdot \chi (X)\), with equality if \(X\) is locally compact; and \(\pi w(X) = w(X)\).
Mathematics Subject Classification ID: 22A30
Mathematics Subject Classification ID: 54H10
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 858602
zbMATH Keywords: rectifiable
zbMATH Keywords: homeomorphism
zbMATH Keywords: topological group
zbMATH Keywords: rectifiable spaces
zbMATH Keywords: metrizability
zbMATH Keywords: weight
zbMATH Keywords: character
zbMATH Keywords: density
zbMATH Keywords: compact-covering number
zbMATH Keywords: locally compact
reviewed by: W. W. Comfort
MaRDI profile type: Publication
full work available at URL: 0166-8641(95)00027-5
OpenAlex ID: W4212921653
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description: scientific article; zbMATH DE number 858602