Topology Proceedings 30 No. 1 (2006), pp. 115-128

Closed Discrete Subspaces in First Countable Spaces

Dennis K. Burke and Kerry D. Smith

In this note, we consider the following three topological statements: (1) NS: In every normal first countable space, closed discrete subspaces are separated; (2) CPS: In every countably paracompact first countable T₂ space, closed discrete subspaces are separated; (3) CMGD: In every countably metacompact first countable T₁ space, closed discrete subspaces are G_d-sets. We construct general combinatorial statements for each of the topological statements above. We then use the combinatorics to prove a chain of implications between the topological statements. We finish by asking if any of the implications can be reversed.

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Topology Proceedings, Volume 30, No. 1 (2006)
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