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.
