Vol. 35, No. 2, 1970

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A necessary and sufficient condition for the embedding of a Lindelof space in a Hausdorff 𝒦σ space

Brenda MacGibbon

Vol. 35 (1970), No. 2, 459–465

It is known that complete regularity characterizes the Hausdorff topological spaces which are embeddable in a compact Hausdorff space. The theory of 𝒦-analytic and 𝒦-Borelian sets leads naturally to the search for an analogous criterion for the embedding of a Hausdoff space in a Hausdorff 𝒦σ space. (A Hausdorff 𝒦σ space is a Hausdorff space which is equal to a countable union of its compact subsets.) We shall give an answer to this problem for Lindelof spaces.

Strong regularity and strong normality of a closed subspace with respect to a given Hausdorff space are defined. It is shown that a Hausdorff Lindelof space is embeddable in a Hausdorff 𝒦σ if and only if X is equal to a union of an increasing sequence of its strongly regular closed subspaces. An example is given of a nonregular space which is equal to a union of an increasing sequence of its strongly normal subspaces.

Mathematical Subject Classification
Primary: 54.40
Received: 14 January 1970
Published: 1 November 1970
Brenda MacGibbon