We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment of X such that d(f(x),f(y)) < ε whenever . We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric space.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-3-2,
author = {Ahmed Bouziad},
title = {The point of continuity property, neighbourhood assignments and filter convergences},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {225-242},
zbl = {1255.54006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-3-2}
}
Ahmed Bouziad. The point of continuity property, neighbourhood assignments and filter convergences. Fundamenta Mathematicae, Tome 219 (2012) pp. 225-242. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-3-2/