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

Vitalij A. Chatyrko

Commentationes Mathematicae Universitatis Carolinae (2021)

  • Volume: 62, Issue: 4, page 483-489
  • ISSN: 0010-2628

Abstract

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In this paper the following two propositions are proved: (a) If X α , α ∈ A , is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product □ α ∈ A X α can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If X α , α ∈ A , is an infinite system of Brown Hausdorff topological spaces then the box product □ α ∈ A X α 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.

How to cite

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Chatyrko, Vitalij A.. "The (dis)connectedness of products of Hausdorff spaces in the box topology." Commentationes Mathematicae Universitatis Carolinae 62.4 (2021): 483-489. <http://eudml.org/doc/297912>.

@article{Chatyrko2021,
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 $\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.},
author = {Chatyrko, Vitalij A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {box topology; connectedness; completely Hausdorff space; Urysohn space; Brown space},
language = {eng},
number = {4},
pages = {483-489},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The (dis)connectedness of products of Hausdorff spaces in the box topology},
url = {http://eudml.org/doc/297912},
volume = {62},
year = {2021},
}

TY - JOUR
AU - Chatyrko, Vitalij A.
TI - The (dis)connectedness of products of Hausdorff spaces in the box topology
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62
IS - 4
SP - 483
EP - 489
AB - 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.
LA - eng
KW - box topology; connectedness; completely Hausdorff space; Urysohn space; Brown space
UR - http://eudml.org/doc/297912
ER -

References

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