Metrizability of Pixley-Roy Hyperspaces

Metrizability of Decomposition Spaces

Roy 27_7

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Simultaneous Metrizability of Coarse Spaces

A metric space can be naturally endowed with both a topology and a coarse structure. We examine the converse to this. Given a topology and a coarse structure we give necessary and sufficient conditions for the existence of a metric giving rise to both of these. We conclude with an application to the construction of the coarse assembly map.

Resolvable Maps Preserve Complete Metrizability

Let X be a Polish space, Y a separable metrizable space, and f : X → Y a continuous surjection. We prove that if the image under f of every open set or every closed set is resolvable, then Y is Polish. This generalizes similar results by Sierpiński, Vainštain, and Ostrovsky.

Remarks on Metrizability

In connection with those paragraphs of my paper Applications of the theory of Boolean rings to general topology (Trans. Amer. Math. Soc. vol. 41 (1937) pp. 375-481) dealing with regular spaces, I have long been curious to know whether certain results proved there could be used to obtain the well known theorem that a separable (Hausdorff) space is metrizable if (and only if) it is regular. Since...