Vol. 36, No. 1, 1971

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Conditions for countable bases in spaces of countable and point-countable type

David Michael Henry

Vol. 36 (1971), No. 1, 181–183

A space X is of countable type if for every compact C X, there exists a compact set K having a countable basis with C K.X is of point-countable type if there exists a covering of compact subsets of X, each having a countable basis. It is shown that in a Hausdorff space of countable type, a compact set has a countable basis if and only if it is a Gδ-set. Similarly, for Hausdorff spaces of point-countable type, a point has a countable basis if and only if it is a Gδ-set.

Mathematical Subject Classification
Primary: 54.40
Received: 4 May 1970
Revised: 26 June 1970
Published: 1 January 1971
David Michael Henry