Geodesic
The (dis)connectedness of products of Hausdorff spaces in the box topology
Commentationes Mathematicae Universitatis Carolinae, Tome 62 (2021) no. 4, pp. 483-489.

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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 $\square_{\alpha \in A} X_\alpha$ 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 $\square_{\alpha \in A} X_\alpha$ 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.
DOI : 10.14712/1213-7243.2022.001
Classification : 54B10, 54D05, 54D10
Keywords: box topology; connectedness; completely Hausdorff space; Urysohn space; Brown space 
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Chatyrko, Vitalij A. The (dis)connectedness of products of Hausdorff  spaces in the box topology. Commentationes Mathematicae Universitatis Carolinae, Tome 62 (2021) no. 4, pp. 483-489. doi : 10.14712/1213-7243.2022.001. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2022.001/

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