|Conţinutul numărului revistei|
| SM ISO690:2012|
CHOBAN, Mitrofan; KENDEROV, Petar; REVALSKI, Julian Petrov. Fragmentability of open sets and topological games. In: Topology and its Applications. 2020, nr. 275, p. 0. ISSN 0166-8641.
|Topology and its Applications|
|Numărul 275 / 2020 / ISSN 0166-8641|
We give internal and external characterizations of the topological spaces X in which there exists a metric d (not necessarily generating the topology of X) which fragments the nonempty open sets. I.e. every nonempty open subset contains open subsets of arbitrarily small d-diameter. One of the characterizations says that every such space X is the union of two disjoint sets X1 and X2 where X1 is of the first Baire category in X and X2 admits a continuous one-to-one mapping into a metrizable space. An external characterization is obtained via the existence of a winning strategy for one of the players in a topological game similar to the Banach-Mazur game. An example of a space X is exhibited for which neither of the players in this game has a winning strategy. Of special interest is the case when the metric d which fragments the nonempty open sets of X is complete and generates a topology finer than the topology of X. This happens if and only if X contains a dense completely metrizable subset.
Baire category, Dense completely metrizable subset, Fragmentability of sets, Topological game, Winning strategy