โ Scribed by A. Makkouk
No coin nor oath required. For personal study only.
๐ SIMILAR VOLUMES
Let X and Y be Hausdorff topological spaces. Let P be the family of all partial maps from X to Y: a partial map is a pair (B,f). where B E CL(X) (= the family of all nonempty closed subsets of X) and f is a continuous function from B to El'. Denote by 7~ the generalized compact-open topology on P.
We prove that for every separable, O-dimensional me&able space X without isolated points, such that every compact subset of it is scattered, the cocompact topology on the hyperspace of X does not coincide with the upper Kuratowski topology-that is, X is dissonant. In particular, it follows that the
We prove that for a metrizable space X the following are equivalent: (i) the free Abelian topological group A(X) is the inductive limit of the sequence {A n (X): n โ N}, where A n (X) is formed by all words of reduced length n; (ii) X is locally compact and the set of all non-isolated points of X is