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Journal Articles Comptes rendus de l'Académie des sciences. Série I, Mathématique Year : 1995

Images usco-compactes des espaces Čech-complets de Lindelöf

Ahmed Bouziad
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Jean Calbrix
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Abstract

It is known that the class of $K$-analytic spaces is stable under usco-compact images in the class of Hausdorff spaces. In this article the authors examine the stability of the class of Čech-complete and Lindelöf spaces under usco-compact images among the $T_1$ completely regular ones. It is shown that Čech-complete Lindelöf spaces are stable only in the class of regular $q$-spaces. In fact, for regular $q$-spaces the following properties are proved to be equivalent: (1) $X$ is Čech-complete and Lindelöf; (2) $(\scr K(X),\tau_v)$ is Čech-complete and Lindelöf; (3) $X$ is a full usco-compact image of a Čech-complete and Lindelöf space; (4) $X$ is a compact-covering (or semiproper) image of a Čech-complete and Lindelöf space. We recall that here $\scr K(X)$ denotes the family of compact subsets of $X$ and $\tau_v$ the Vietoris topology on it; a map $\phi$ from a topological space $Y$ to $\scr K(X)$ is said to be full if $\phi(Y)$ is cofinal in $(\scr K(X),\subset)$; $f\colon X\to Y$ is compact-covering or semiproper if every compact subset of $Y$ is the image of a compact subset of $X$
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Dates and versions

hal-00373447 , version 1 (05-04-2009)

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  • HAL Id : hal-00373447 , version 1

Cite

Ahmed Bouziad, Jean Calbrix. Images usco-compactes des espaces Čech-complets de Lindelöf. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 1995, 320 (7), pp.839--842. ⟨hal-00373447⟩
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