On Gibson functions with connected graphs

Year, pages: 2013, 479 - 492

A function $f: X \to Y$ between topological spaces is said to be a weakly Gibson function if $f(\overline{G})\subseteq \overline{f(G)}$ for any open connected set $G\subseteq X$. We call a function $f: X \to Y$ segmentary connected if $X$ is topological vector space and $f([a,b])$ is connected for every segment $[a,b]\subseteq X$. We show that if $X$ is a hereditarily Baire space, $Y$ is a metric space, $f: X \to Y$ is a Baire-one function and one of the following conditions holds: (i) $X$ is a connected and locally connected space and $f$ is a weakly Gibson function, (ii) $X$ is an arcwise connected space and $f$ is a Darboux function, (iii) $X$ is a topological vector space and $f$ is a segmentary connected function, then $f$ has a connected graph.

Karlova, O., Mykhaylyuk, V. 2013. On Gibson functions with connected graphs. In Mathematica Slovaca, vol. 63, no.3, pp. 479-492. 0139-9918. DOI: https://doi.org/10.2478/s12175-013-0110-9

Karlova, O., Mykhaylyuk, V. (2013). On Gibson functions with connected graphs. Mathematica Slovaca, 63(3), 479-492. 0139-9918. DOI: https://doi.org/10.2478/s12175-013-0110-9