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True preimages of compact or separable sets for functional analysts. (English) Zbl 07217159

In this paper, some results are given about existence of true preimages under continuous open maps between \(F\)-spaces, \(F\)-lattices, and some other spaces.
Section 1, with an introductory character, is devoted to the statement of the main result in the paper.
{Theorem 1.} Let \(f\) be a continuous linear operator from an \(F\)-space \(X\) onto an \(F\)-space \(Y\). Then,
1)
Every compact set \(C\) of \(Y\) has a compact preimage \(K\) in \(X\).
2)
If \(X\) is a locally convex \(F\)-space (i.e., a Fréchet space) and \(C\) is (in addition) convex or absolutely convex, then the preimage subset \(K\) may be also chosen of the precise type.

Then, in addition to this, two related results are given (either of these implying the main result, modulo the open mapping theorem):
{Theorem 2.} Let \(f\) be a continuous open map from a complete metric space \(X\) onto a Hausdorff space \(Y\). Then, every compact subset of \(Y\) has a compact preimage in \(X\).
{Theorem 3.} Let \(X\) be a complete metric space, \(Y\) be a Hausdorff space, and \(F:X\to Y\) be a lower semicontinuous set-valued map with nonempty closed values. Then, for every compact subset \(C\) of \(Y\) there is a compact subset \(K\) of \(X\) such that \(C\subset F^{-1}(K)\).
In Section 2, a proof of these results is performed, under compact preimages; and, in Section 3, analogous results are given in the context of separable preimages. Further, in Section 4, the setting is moved to the case of \(F\)-lattices; and, in Section 5, the finite dimensional case is considered.
Further aspects suggested by these developments are also discussed.

MSC:

46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
54E45 Compact (locally compact) metric spaces
46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
54E35 Metric spaces, metrizability
54E40 Special maps on metric spaces
54E50 Complete metric spaces
54D65 Separability of topological spaces
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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References:

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