A characterization of spaces \(l\)-equivalent to the unit interval.
type: Item
instance of: scholarly article
title: A characterization of spaces \(l\)-equivalent to the unit interval.
zbMATH Open document ID: 1053.54024
author: Tamás Mátrai
published in: Topology and its Applications
publication date: 2004-03-14T00:00:00Z
review text: First of all, let us quote the author's abstract: ``Two topological spaces \(X\) and \(Y\) are said to be \(\ell\)-equivalent if there is a linear homeomorphism between \(C_p(X)\) and \(C_p(Y)\), the spaces of continuous functions on \(X\) and \(Y\) endowed with the topology of pointwise convergence. We prove a necessary condition on a topological space to be \(\ell\)-equivalent to the \(n\)-dimensional cube \(I^n\).'' The author proves the following theorems. Definition 5. Let \(X\) be a topological space. For an \(n\in\mathbb{N}\), let \[ I_n(X)= \bigcup_{s\in S} U_s,\quad J_n(X)= \bigcup_{t\in T} V_t, \] where \(\{U_s : s\in S\}\) is the family of all open subsets of \(X\) which can be embedded into \(I^n\) and \(\{V_t: t\in T\}\) is the family of all open subsets of \(X\) which are at most \((n- 1)\)-dimensional. For an ordinal \(\alpha\), the \(\alpha\)th \(n\)-dimensional embedding derivative \(X^{[\alpha,n]}\) is defined in the following way: \[ \begin{gathered} X^{[0,n]}= X;\\ X^{[\alpha+ 1,n]}= X^{[\alpha,n]}\setminus I_n(X^{[\alpha, n]});\\ X^{[\alpha,n]}= \bigcap_{\beta< \alpha} X^{[\beta,n]}\text{ if \(\alpha\) is limit}.\end{gathered} \] Similarly, the \(\alpha\)th \(n\)-dimensional strong embedding derivative \(X^{\alpha,n}\) is defined in the following way: \[ \begin{gathered} X^{0,n}= X\setminus J_n(X);\\ X^{\alpha+ 1,n}= X^{\alpha, n}\setminus (I_n(X^{\alpha, n})\cup J_n(X^{\alpha,n}\setminus I_n(X^{\alpha, n})));\\ X^{\alpha, n}= \bigcup_{\beta< \alpha} X^{\beta, n}\setminus J_n\Biggl(\bigcap_{\beta< \alpha} X^{\beta, n}\Biggr)\text{ if \(\alpha\) is limit}.\end{gathered} \] Theorem 6. If \(X\) is \(\ell\)-equivalent to \(I^n\), then \(X^{m,n}=\emptyset\) for some \(m\in\mathbb{N}\). For \(n= 1\) and a one-dimensional compact metrizable \(X\), this condition is sufficient. Theorem 18. Let \(X\) be an \(n\)-dimensional compact metric space, and let \[ H: C_p(X)\to C_p(X) \] be a linear homeomorphism. If \(…
reviewed by: Akihiro Okuyama
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 54F65
zbMATH DE Number: 2056834
zbMATH Keywords: function spaces
zbMATH Keywords: topology of pointwise convergence
zbMATH Keywords: linear homeomorphisms
zbMATH Keywords: \(l\)-equivalence
MaRDI profile type: Publication
full work available at URL: j.topol.2003.08.017
OpenAlex ID: W2069927817
cites work: Q3736526
cites work: Q3672683
cites work: Q4000501
cites work: An Example of l p -Equivalent Spaces Which Are Not l ∗ p -Equivalent
cites work: Q4185470
cites work: Q4293580
cites work: Q3667604
cites work: A characterization of compactness in terms of uniform structure in a function space
DOI: 10.1016/J.TOPOL.2003.08.017
description: scientific article; zbMATH DE number 2056834