On weakly Gibson $F_{\sigma} $-measurable mappings

Olena Karlova¹ ; Volodymyr Mykhaylyuk¹

¹ Chernivtsi National University Department of Mathematical Analysis 58012 Chernivtsi, Ukraine

Colloquium Mathematicum, Tome 133 (2013) no. 2, pp. 211-219.

Résumé

A function $f:X\to Y$ between topological spaces is said to be a weakly Gibson function if $f(\overline {U})\subseteq \overline {f(U)}$ for any open connected set $U\subseteq X$. We prove that if $X$ is a locally connected hereditarily Baire space and $Y$ is a $T_1$-space then an $F_\sigma $-measurable mapping $f:X\to Y$ is weakly Gibson if and only if for any connected set $C\subseteq X$ with dense connected interior the image $f(C)$ is connected. Moreover, we show that each weakly Gibson $F_\sigma $-measurable mapping $f:\mathbb R^n\to Y$, where $Y$ is a $T_1$-space, has a connected graph.

Détail

DOI : 10.4064/cm133-2-7

Keywords: function between topological spaces said weakly gibson function overline subseteq overline connected set subseteq prove locally connected hereditarily baire space space sigma measurable mapping weakly gibson only connected set subseteq dense connected interior image connected moreover each weakly gibson sigma measurable mapping mathbb where space has connected graph