A characterization of open mapping in terms of convergent sequences

It is certainly well known that a mapping between metric spaces is continuous if and only if it preserves convergent sequences. Does there exist a comparable characterization for the mapping to be open? Of course, the inverse mapping is set-valued, in general. In this research/expository note, we show that a mapping is open if and only if the set-valued inverse mapping preserves convergent sequences in an appropriate set-theoretic sense

A characterization of open mapping in terms of convergent sequences

It is certainly well known that a mapping between metric spaces is continuous if and only if it preserves convergent sequences. Does there exist a comparable characterization for the mapping to be open? Of course, the inverse mapping is set-valued, in general. In this research/expository note, we show that a mapping is open if and only if the set-valued inverse mapping preserves convergent sequences in an appropriate set-theoretic sense

A characterization of open mapping in terms of convergent sequences

It is certainly well known that a mapping between metric spaces is continuous if and only if it preserves convergent sequences. Does there exist a comparable characterization for the mapping to be open? Of course, the inverse mapping is set-valued, in general. In this research/expository note, we show that a mapping is open if and only if the set-valued inverse mapping preserves convergent sequences in an appropriate set-theoretic sense

Stationary and convergent strategies in Choquet games

If NONEMPTY has a winning strategy against EMPTY in the Choquet game on a space, the space is said to be a Choquet space. Such a winning strategy allows NONEMPTY to consider the entire finite history of previous moves before making each new move; a stationary strategy only permits NONEMPTY to consider the previous move by EMPTY. We show that NONEMPTY has a stationary winning strategy for every second countable T1 Choquet space. More generally, NONEMPTY has a stationary winning strategy for any T1 Choquet space with an open-finite basis. We also study convergent strategies for the Choquet game, proving the following results. (1) A T1 space X is the open image of a complete metric space if and only if NONEMPTY has a convergent winning strategy in the Choquet game on X. (2) A T1 space X is the compact open image of a metric space if and only if X is metacompact and NONEMPTY has a stationary convergent strategy in the Choquet game on X. (3) A T1 space X is the compact open image of a complete metric space if and only if X is metacompact and NONEMPTY has a stationary convergent winning strategy in the Choquet game on X.Comment: 24 page

Beyond Topologies, Part I

Arguments on the need, and usefulness, of going beyond the usual Hausdorff-Kuratowski-Bourbaki, or in short, HKB concept of topology are presented. The motivation comes, among others, from well known {\it topological type processes}, or in short TTP-s, in the theories of Measure, Integration and Ordered Spaces. These TTP-s, as shown by the classical characterization given by the {\it four Moore-Smith conditions}, can {\it no longer} be incorporated within the usual HKB topologies. One of the most successful recent ways to go beyond HKB topologies is that developed in Beattie & Butzmann. It is shown in this work how that extended concept of topology is a {\it particular} case of the earlier one suggested and used by the first author in the study of generalized solutions of large classes of nonlinear partial differential equations