D-space

In mathematics, a topological space $X$ is a D-space if for any family $\{U_{x}:x\in X\}$ of open sets such that $x\in U_{x}$ for all points $x\in X$ , there is a closed discrete subset $D$ of the space $X$ such that $\bigcup _{x\in D}U_{x}=X$ .

History

The notion of D-spaces was introduced by Eric Karel van Douwen and E.A. Michael. It first appeared in a 1979 paper by van Douwen and Washek Frantisek Pfeffer in the Pacific Journal of Mathematics.[1] Whether every Lindelöf and regular topological space is a D-space is known as the D-space problem. This problem is among twenty of the most important problems of set theoretic topology.[2]

Properties

Every Menger space is a D-space[3]
A subspace of a topological linearly ordered space is a D-space iff it is a paracompact space.[4]

References

van Douwen, E.; Pfeffer, W. (1979). "Some properties of the Sorgenfrey line and related spaces" (PDF). Pacific Journal of Mathematics. 81: 371–377.
Elliott., Pearl (2007-01-01). Open problems in topology II. Elsevier. ISBN 9780444522085. OCLC 162136062.
Aurichi, Leandro (2010). "D-Spaces, Topological Games, and Selection Principles" (PDF). Topology Proceedings. 36: 107–122.
van Douwen, Eric; Lutzer, David (1997-01-01). "A note on paracompactness in generalized ordered spaces". Proceedings of the American Mathematical Society. 125 (4): 1237–1245. doi:10.1090/S0002-9939-97-03902-6. ISSN 0002-9939.

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[1] van Douwen, E.; Pfeffer, W. (1979). "Some properties of the Sorgenfrey line and related spaces" (PDF). Pacific Journal of Mathematics. 81: 371–377.

[2] Elliott., Pearl (2007-01-01). Open problems in topology II. Elsevier. ISBN 9780444522085. OCLC 162136062.

[3] Aurichi, Leandro (2010). "D-Spaces, Topological Games, and Selection Principles" (PDF). Topology Proceedings. 36: 107–122.

[4] van Douwen, Eric; Lutzer, David (1997-01-01). "A note on paracompactness in generalized ordered spaces". Proceedings of the American Mathematical Society. 125 (4): 1237–1245. doi:10.1090/S0002-9939-97-03902-6. ISSN 0002-9939.

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