Local properties and maximal Tychonoff connected spaces

Abstract

We prove that, if $X$ is a Tychonoff connected space and $\chi (x, X) \leq \omega$ for some $x \in X$, then there exists a strictly stronger Tychonoff connected topology on the space $X$, i.e., the space $X$ is not maximal Tychonoff connected. We also establish that if X is locally connected or CT-compact or has pointwise countable type then $X$ cannot be maximal Tychonoff connected.