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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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