## Vitalij A. Chatyrko

*The (dis)connectedness of products of Hausdorff spaces in the box topology*

Comment.Math.Univ.Carolin. 62,4 (2021) 483-489.**Abstract: **In this paper the following two propositions are proved: (a) If X_\alpha, \alpha \in A, is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product {\mathlarger{\mathlarger{\mathlarger\square}}}_{\alpha \in A} X_\alpha can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If X_\alpha, \alpha \in A, is an infinite system of Brown Hausdorff topological spaces then the box product {\mathlarger{\mathlarger{\mathlarger\square}}}_{\alpha \in A} X_\alpha is also Brown Hausdorff, and hence, it is connected. A space is Brown if for every pair of its open nonempty subsets there exists a point common to their closures. There are many examples of countable Brown Hausdorff spaces in literature.

**Keywords:** box topology; connectedness; completely Hausdorff space; Urysohn space; Brown space

**DOI:** DOI 10.14712/1213-7243.2022.001

**AMS Subject Classification:** 54B10 54D05 54D10

PDF