Function Spaces with Uniform, Fine and Graph Topologies by Robert A. McCoy Subiman Kundu & Varun Jindal

Author:Robert A. McCoy, Subiman Kundu & Varun Jindal
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Now we define a generalization of compact spaces. We say that a topological space X is generalized compact or GK if every open cover of X has a subcover such that . Also for a topological space X, the extent, e(X) of X is defined as

Proposition 3.8

([17], Theorem 7) A metrizable space X attains its extent if and only if it is not GK.

Also, for arbitrary sets A and B, we will denote by the set of all functions from B to A.

For any two cardinals , we have . In an analogous way, we write and . In particular, the expression means that , and is clearly equal to whenever is a non-successor cardinal.

Lemma 3.2

([24], Proposition 2.5) Let D be closed and discrete subset of a metrizable space X, let and E be an -uniformly discrete subset of a pathwise connected metric space . Then there exists an -uniformly discrete subset of C(X, Y) of cardinality .

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