On the Multiplicity of Jigsawed Bases in Compact and Countably Compact Spaces
by
John T. Griesmer
Miami University
Coauthors: Zoltan T. Balogh (Miami University)

A collection of subsets P of a set X is said to be finite-in-countable if for all A in [X]^\omega, there is a finite B subset A such that B subset P for only countably many P in P.

A collection P of subsets of a topological space X is called a jigsawed base of neighborhoods (resp. jigsawed base) if for all x in X, for all open U with x in U, there exists F in [P] ^{< \omega} with x in ( \cup F) ^o = \cup F subset U (resp. x in ( \cup F) ^o subset \cup F subset U).

We prove several theorems concerning compact and countably compact spaces with finite-in-countable jigsawed bases. Every compact space with a finite-in-countable jigsawed base is metrizable. We extend this result to the case of countable compactness.

A collection P of subsets of X is called \omega-in-countable if for all A in [X]^\omega, A subset P for only countably many P in P. We prove that The claim every compact space with an \omega-in-countable jigsawed base is metrizable" is independent of ZFC, is decided false by CH, and is decided true by \wp > \omega₁.

Date received: February 22, 2002