Injections into function spaces over ordinals
type: Item
instance of: scholarly article
title: Injections into function spaces over ordinals
zbMATH Open document ID: 1211.54031
author: Raushan Z. Buzyakova
published in: Topology and its Applications
publication date: 2011-01-12T00:00:00Z
review text: For topological spaces \(X\), \(Y\), \(Z\) let \(C_p(X, Z)\) be the space of continuous mappings from \(X\) into \(Z\) with the topology of pointwise convergence. The present paper is one of several recent studies of the relation between spaces \(X\), \(Y\) of ordinals and the presence of some sort of relation between \(C_p(X, Z)\) and \(C_p(Y, Z)\). The case where \(Z\) is a finite cyclic group was studied by \textit{L. V. Genze}, \textit{S. P. Gul'ko} and \textit{T. E. Khmyleva} [Topol. Proc. 38, 1--15 (2011; Zbl 1210.54034)]. The case where \(Z\) is a metric space with two or more points is studied in the present paper. The author shows, among other things, that if \(\tau\) is a regular uncountable cardinal, \(X\subset\tau\) is unbounded, and there is a continuous injection from \(C_p(X, Z)\) into \(C_p(T, Z)\), then \(X\) contains a closed subspace homeomorphic to \(\tau\). Her main result says that if \(X\subset\omega_1\), then the following are equivalent: (i) \(X\) is either countable or contains a homeomorphic copy of \(\omega_1\); (ii) there is a continuous injection from \(C_p(X, Z^\omega)\) into \(C_p(\omega_1, Z^\omega)\). Some open questions are also raised.
reviewed by: James V. Whittaker
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 03E10
zbMATH DE Number: 5835564
zbMATH Keywords: embedding of function spaces
zbMATH Keywords: stationary set of ordinals
MaRDI profile type: Publication
full work available at URL: j.topol.2010.09.006
OpenAlex ID: W2034592958
cites work: Q4228099
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DOI: 10.1016/J.TOPOL.2010.09.006
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description: scientific article; zbMATH DE number 5835564