On Gibson functions with connected graphs

TL;DR

This paper investigates conditions under which certain classes of functions between topological spaces have connected graphs, focusing on weakly Gibson, Darboux, and segmentary connected functions in various topological contexts.

Contribution

It establishes that functions satisfying specific Gibson, Darboux, or segmentary connectedness conditions have connected graphs under certain topological assumptions.

Findings

Weakly Gibson functions with connected graphs in hereditarily Baire spaces.

Darboux functions with connected graphs in arcwise connected spaces.

Segmentary connected functions with connected graphs in topological vector spaces.

Abstract

A function $f:X\to Y$ between topological spaces is said to be a {\it weakly Gibson function} if $f(\overline{G})\subseteq \overline{f(G)}$ for any open connected set \mbox{$G\subseteq X$}. We call a function $f:X\to Y$ {\it 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, \mbox{$f:X\to Y$} 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.