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# $D$-spaces and resolution

Published:1997-12-01
Printed: Dec 1997
• Zineddine Boudhraa
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## Abstract

A space $X$ is a $D$-space if, for every neighborhood assignment $f$ there is a closed discrete set $D$ such that $\bigcup{f(D)}=X$. In this paper we give some necessary conditions and some sufficient conditions for a resolution of a topological space to be a $D$-space. In particular, if a space $X$ is resolved at each $x\in X$ into a $D$-space $Y_x$ by continuous mappings $f_x\colon X-\{{x}\} \rightarrow Y_x$, then the resolution is a $D$-space if and only if $\bigcup{\{{x}\}}\times \Bd(Y_x)$ is a $D$-space.
 Keywords: $D$-space, neighborhood assignment, resolution, boundary
 MSC Classifications: 54D20 - Noncompact covering properties (paracompact, Lindelof, etc.) 54B99 - None of the above, but in this section 54D10 - Lower separation axioms ($T_0$--$T_3$, etc.) 54D30 - Compactness

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