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Solution of Ponomarev’s problem of condensation onto compact sets. (English. Russian original) Zbl 1467.54002

Sib. Math. J. 62, No. 1, 131-137 (2021); translation from Sib. Mat. Zh. 62, No. 1, 164-172 (2021).
A condensation is a continuous bijection between topological spaces, or, if one identifies domain and image, passing to a smaller topology on the same set. The main question, due to Alexandroff, is when a Hausdorff space has a condensation to a compact Hausdorff space. A related problem is which compact Hausdorff spaces are \(a\)-spaces, which means that every co-countable subspace has a condensation to a compact Hausdorff space. Compact metric spaces are \(a\)-spaces, as are compact first-countable zero-dimensional and compact ordered spaces. Ponomarev asked whether perfectly normal compact spaces are \(a\)-spaces. In [E. G. Pytkeev, Sov. Math., Dokl. 26, 162–165 (1982; Zbl 0529.54009); translation from Dokl. Akad. Nauk SSSR 265, 819–823 (1982)] the second author showed that first-countability (points are \(G_\delta\)) does not suffice. The present paper is devoted to the construction of a counterexample to Ponomarev’s full question, under the assumption of the Continuum Hypothesis.
Reviewer: K. P. Hart (Delft)

MSC:

54C10 Special maps on topological spaces (open, closed, perfect, etc.)
03E50 Continuum hypothesis and Martin’s axiom
54A35 Consistency and independence results in general topology
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D30 Compactness
54G20 Counterexamples in general topology

Citations:

Zbl 0529.54009
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References:

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