On weakly Gibson $F_{\sigma} $-measurable mappings
Volume 133 / 2013
Colloquium Mathematicum 133 (2013), 211-219
MSC: Primary 26B05, 54C08; Secondary 26A21.
DOI: 10.4064/cm133-2-7
Abstract
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.
