Tree sums of maximal connected spaces

Abstract

A topology $\tau$ on a set $X$ is called maximal connected if it is connected, but no strictly finer topology ${\tau }^{\ast }>\tau$ is connected. We consider a construction of so-called tree sums of topological spaces, and we show how this construction preserves maximal connectedness and also related properties of strong connectedness and essential connectedness. We also recall the characterization of finitely generated maximal connected spaces and reformulate it in the language of specialization preorder and graphs, from which it is clear that finitely generated maximal connected spaces are precisely $T_{\frac{1}{2}}$ tree sums of copies of the Sierpiński space.

arXiv comment29 pages, 3 figures, final version