The continuity of the map lim_T in Hausdorff spaces

Details The continuity of the map lim_T in Hausdorff spaces The continuity of the map lim_T in Hausdorff spaces

2004-07-21

arxiv.org/abs/math/0407362v1

Mathematics

General Topology

LaTeXe-document. Uses gtart.cls, amsmath, amssymb, xy and babel with british option. Number of pages 11, no figures

Scientific paper

Consider a Hausdorff space (X,T) and a set C of converging nets in X. By virtue of the limit uniqueness, the relation Lim which assigns each member x of X to every net N lying in C that converges to x is a map. Of course, structuring C with some topology U, Lim can be a continuous map. If T is a topology induced by a uniformity, and F is a function space such that X is the codomain of each f in F, it is a well-known property the uniform limits of continuous functions to be continuous. In this paper, the author shows that the continuity of limits of continuous-map nets is implied by the continuity of the map Lim, whenever the involved topologies are large enough, being this result obtained without using neither the uniform convergence notion nor the uniformity concept.

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