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Bijections onto compact spaces. (English) Zbl 0676.54028
The question of when a Hausdorff space has a continuous bijection onto a compact Hausdorff space is attributed to Alexandroff. In 1972, V. I. Belugin [Dokl. Akad. Nauk SSSR 207, 259-261 (1972; Zbl 0269.54013)] showed that every cocountable subspace of a dyadic space has such a bijection, but Ponomarev showed that no cocountable subspace of \(\beta\) \(\omega\)-\(\omega\) does.
Let T be an infinite set, \(S\subseteq T\); let \(D^ T=\{p| p:\) \(T\to \{0,1\}\}\), and for \(p\in D^ T\), let \(G_ S(p)=\{f\in D^ T:\) \(f| S=p| S\), and \(p^{-1}(0)\subseteq f^{-1}(0)\}\). A subset X of \(D^ T\) is called an \(\omega\)-set if, for each \(p\in X\), there exists \(S\subseteq T\) with \(| S| \leq \omega\) and \(G_ S(p)\subseteq X\). A space Y is weakly dyadic if Y is a continuous image of a compact \(\omega\)-set in \(D^ T.\)
It is then shown that any cocountable subspace of a weakly dyadic space has a continuous bijection onto a compact Hausdorff space. The weakly dyadic spaces properly include the centred spaces of M. G. Bell [Fundam. Math. 125, 47-58 (1985; Zbl 0589.54019)].
Reviewer: D.L.Grant
MSC:
54D30 Compactness
54C05 Continuous maps
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