# Tree sums of maximal connected spaces

## TL;DR

This paper investigates the properties of maximal connected topologies, introduces a tree sum construction that preserves such properties, and characterizes finitely generated maximal connected spaces using preorder and graph concepts.

## Contribution

It introduces the tree sum construction for topological spaces and demonstrates its preservation of maximal connectedness and related properties, providing a new characterization of finitely generated maximal connected spaces.

## Key findings

- Tree sums preserve maximal connectedness.
- Finitely generated maximal connected spaces are $T_{1/2}$ tree sums of Sierpiński spaces.
- Characterization of these spaces via specialization preorder and graphs.

## Abstract

A topology $\tau$ on a set $X$ is called maximal connected if it is connected, but no strictly finer topology $\tau^* > \tau$ 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 Sierpi\'nski space.