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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



When do connected spaces have nice connected preimages?

Author: Vladimir V. Tkachuk
Journal: Proc. Amer. Math. Soc. 126 (1998), 3437-3446
MSC (1991): Primary 54A25
MathSciNet review: 1459154
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Abstract: We prove that every connected Tychonoff space is an open monotone continuous image of a connected strictly $\sigma$-discrete left-separated Tychonoff space. For wide classes of connected spaces it is established that they have a finer Hausdorff strictly $\sigma$-discrete connected topology. Another result is that a finer Tychonoff connected strictly $\sigma$-discrete topology exists for any Tychonoff topology with a countable network. We show that there are Tychonoff connected spaces with countable network which are not continuous images of connected second countable spaces. It is established also that every connected Tychonoff space $\mathcal {X}$ is an open retract of a connected homogeneous Tychonoff space, while it is not always possible to find a finer connected homogeneous topology on $\mathcal {X}$.

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Vladimir V. Tkachuk
Affiliation: Departamento de Matematicas, Universidad Autónoma Metropolitana, Av. Michoacan y La Purísima, Iztapalapa, A.P. 55-532, C.P. 09340, Mexico, D.F.

Keywords: Connected space, metric space, open map, preimage, finer connected topology, homogeneous space
Received by editor(s): November 14, 1996
Received by editor(s) in revised form: April 4, 1997
Communicated by: Alan Dow
Article copyright: © Copyright 1998 American Mathematical Society