On a Question of Closed Maps of S. Lin

• Journal title : Kyungpook mathematical journal
• Volume 50, Issue 4,  2010, pp.537-543
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2010.50.4.537
Title & Authors
On a Question of Closed Maps of S. Lin
Chen, Huaipeng;

Abstract
Let X be a regular $\small{T_1}$-space such that each single point set is a $\small{G_{\delta}}$ set. Denot 'hereditarily closure-preserving' by 'HCP'. To consider a question of closed maps of S. Lin in [6], we improve some results of Foged in [1], and prove the following propositions. Proposition 1. $\small{D\;=\;\{x{\in}X\;:\;\mid\{F{\in}\cal{F}:x{\in}F\}\mid{\geq}{\aleph}_0\}}$ is discrete and closed if $\small{\cal{F}}$ is a collection of HCP. Proposition 2. $\small{\cal{H}\;=\;\{{\cup}\cal{F}$ is an fininte subcolletion of $\small{\cal{F}_n\}}$ is HCP if $\small{\cal{F}}$ is a collection of HCP. Proposition 3. Let (X,$\small{\tau}$) have a $\small{\sigma}$-HCP k-network. Then (X,$\small{\tau}$) has a $\small{\sigma}$-HCP k-network F = $\small{{\cup}_n\cal{F}_n}$ such that such tat: (i) $\small{\cal{F}_n\;\subset\;\cal{F}_{n+1}}$, (ii) $\small{D_n\;=\;\{x{\in}X\;:\;\mid\{F{\in}\cal{F}_n\;:\;x{\in}F\}\mid\;{\geq}\;{\aleph}_0\}}$ is a discrete closed set and (iii) each $\small{\cal{F}_n}$ is closed to finite intersections.
Keywords
$\small{{\aleph}}$--spaces;k-networks;closed maps;
Language
English
Cited by
References
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