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2008•Advances in MathematicsRequires access

A Special Point-countable Family That Makes a Space to be a D-space

Liang-Xue Peng

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Abstract

In this note,we give a sufficient condition that a space to be a D-space.The following is the main conclusion:If a space X has a point-countable family F,such that for any set A■X,if A is not closed in X,then there exists some point x∈■\A,satisfying that for any open set U of X,suppose x∈U,there exists some F∈F,such that x∈F■U and F∩A≠■.Then X is a D-space.By this result,we will know that a sequential space with a point-countable cs~*-network is a D-space.

About this research paper

What this paper is about

In this note,we give a sufficient condition that a space to be a D-space.The following is the main conclusion:If a space X has a point-countable family F,such that for any set A■X,if A is not closed in X,then there exists some point x∈■\A,satisfying that for any open set U of X,suppose x∈U,there exists some F∈F,such that x∈F■U and F∩A≠■.Then X is a D-space.By this result,we will know that a sequential space with a point-countable cs~*-network is a D-space.

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Available abstract

In this note,we give a sufficient condition that a space to be a D-space.The following is the main conclusion:If a space X has a point-countable family F,such that for any set A■X,if A is not closed in X,then there exists some point x∈■\A,satisfying that for any open set U of X,suppose x∈U,there exists some F∈F,such that x∈F■U and F∩A≠■.Then X is a D-space.By this result,we will know that a sequential space with a point-countable cs~*-network is a D-space.

Key concepts: Countable set, Mathematics, Space (punctuation), Cosmic space, Regular space, Isolated point, Point (geometry), Second-countable space

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