Shrinking in perfect preimages of shrinking spaces
type: Item
instance of: scholarly article
title: Shrinking in perfect preimages of shrinking spaces
zbMATH Open document ID: 0864.54018
DOI: 10.1016/0166-8641(95)00084-4
author: Zorana Lazarevic
published in: Topology and its Applications
publication date: 1997-06-22T00:00:00Z
review text: An open cover \(\{V_\alpha:\alpha\in k\}\) of a space \(X\) is a shrinking for another open cover \(\{U_\alpha : \alpha\in k\}\) if \(\text{cl}(V_\alpha)\subset U_\alpha\) for every \(\alpha\in k\). A space \(X\) is \(k\)-shrinking if every open cover of cardinality \(\leq k\) has a shrinking; \(X\) is shrinking if it is \(k\)-shrinking for every \(k\). It is shown in the first part of the paper that a \(k\)-paracompact perfect preimage of a shrinking space is \(k^+\)-shrinking. As a corollary the author obtained that a normal perfect preimage of a shrinking space is \(\omega_1\)-shrinking. The aim of the next part is to prove that a normal perfect preimage of a generalized ordered space is shrinking. This result is used then to show that a normal perfect preimage of a monotonically normal space is shrinking.
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54E20
Mathematics Subject Classification ID: 54F05
zbMATH DE Number: 937171
zbMATH Keywords: perfect map
zbMATH Keywords: shrinking space
zbMATH Keywords: generalized ordered space
zbMATH Keywords: monotonically normal space
reviewed by: V. Valov
MaRDI profile type: Publication
full work available at URL: 0166-8641(95)00084-4
OpenAlex ID: W1987777509
cites work: Monotone normality
cites work: Normality in products
cites work: Q4150804
cites work: Paracompactness in ordered spaces
cites work: Set theory. An introduction to independence proofs
cites work: Q3925533
description: scientific article; zbMATH DE number 937171