The theory of pursuit games is obviously fragmentary at present. We know that general determinability of such games is incompatible with analysis, based on the principle of time continuity; but we also witness some reasonably successful probing on a smaller scale. The problem is one of existence of winning strategies for quite general sets and spaces. It will be shown here that in one case, where multivalued strategies are used, such strategies must necessarily be subclasses of Polish spaces and in the other, the monovalued case, the loser's set either has to be a first category set in the sense of Baire or an ideal, but in any case a kind of small set. This paper is meant to provide a common topological basis for the appreciation of more recent results.
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