The point of continuity property, neighbourhood assignments and filter convergences
Volume 218 / 2012
Fundamenta Mathematicae 218 (2012), 225-242
MSC: 54C08, 26A21, 28A20, 54E20, 54F05, 54H05.
DOI: 10.4064/fm218-3-2
Abstract
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.
