It is known that complete regularity characterizes the Hausdorff topological spaces which are embeddable in a compact Hausdorff space. The theory of ^^-analytic and J^~-Borelian sets leads naturally to the search for an analogous criterion for the embedding of a Hausdoff space in a Hausdorff 3ZI space. (A Hausdorff J%Γ a 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

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... 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 3ίΓ c 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.