\(C^*\)-embedding, and \(C\)-embedding on product spaces
type: Item
instance of: scholarly article
title: \(C^*\)-embedding, and \(C\)-embedding on product spaces
zbMATH Open document ID: 0887.54017
author: Kaori Yamazaki
published in: Tsukuba Journal of Mathematics
publication date: 1998-05-10T00:00:00Z
review text: A subspace \(A\) of a space \(X\) is said to be \(C\)-embedded (resp. \(C^\ast\)-embedded) in \(X\) if every real-valued, continuous (resp. bounded continuous) function on \(A\) can be extended over \(X\). An interesting problem in extension theory is under what conditions on \(A\) and \(X\) a \(C^\ast\)-embedded closed subspace \(A\) of \(X\) is \(C\)-embedded in \(X\). In particular, the following problem is open: Assume that \(A\) is a \(C\)-embedded closed subspace of a Tychonoff space \(X\), \(Y\) is a space and \(A\times Y\) is \(C^\ast\)-embedded in \(X\times Y\). Then, is \(A\times Y\) \(C\)-embedded in \(X\times Y\)? The author proves that the answer to the problem above is positive if \(Y\) is a \(\sigma\)-locally compact, paracompact, Hausdorff space. Moreover, she proves: (1) If \(A\) is a \(C\)-embedded subspace of a normal \(P\)-space \(X\), \(Y\) is a paracompact, Hausdorff \(\Sigma\)-space and \(A\times Y\) is \(C^\ast\)-embedded in \(X\times Y\), then \(A\times Y\) is \(C\)-embedded in \(X\times Y\); and (2) if \(A\) is a \(P\)-embedded subspace of a normal \(P\)-space \(X\), \(Y\) is a paracompact, Hausdorff \(\sigma\)-space and \(A\times Y\) is \(C^\ast\)-embedded in \(X\times Y\), then \(A\times Y\) is \(P\)-embedded in \(X\times Y\).
Mathematics Subject Classification ID: 54C45
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D45
zbMATH DE Number: 1102904
zbMATH Keywords: \(C^ \ast\)-embedding
zbMATH Keywords: \(C\)-embedding
zbMATH Keywords: product
zbMATH Keywords: \(\sigma\)-locally compact
reviewed by: Haruto Ohta
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163255
OpenAlex ID: W600477473
DOI: 10.21099/TKBJM/1496163255
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description: scientific article; zbMATH DE number 1102904