The point of continuity property, neighbourhood assignments and filter convergences

Ahmed Bouziad¹

¹ Département de Mathématiques Université de Rouen, UMR CNRS 6085 Avenue de l'Université, BP 12 F-76801 Saint-Étienne-du-Rouvray, France

Fundamenta Mathematicae, Tome 218 (2012) no. 3, pp. 225-242

Résumé

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 $f: X\to (Z,d)$ is equivalent to the following condition: $(*)$ For every $\varepsilon>0$, there is a neighbourhood assignment $(V_x)_{x\in X}$ of $X$ such that $d(f(x),f(y))\varepsilon$ whenever $(x,y)\in V_y\times V_x$. We also give various descriptions of the filters $\mathcal F$ on the integers $\mathbb N$ for which ($*$) is satisfied by the $\mathcal F$-limit of any sequence of continuous functions from a topological space into a metric space.

Détail

DOI : 10.4064/fm218-3-2

Keywords: large classes topological spaces metric space point continuity property function equivalent following condition * every varepsilon there neighbourhood assignment varepsilon whenever times various descriptions filters mathcal integers mathbb which * satisfied mathcal f limit sequence continuous functions topological space metric space