Neither first countable nor Cech-complete spaces are maximal Tychonoff connected

Shakhmatov, D.; Tkačenko, M.; Tkachuk, V.; Watson, S.; Wilson, R.

doi:10.1090/S0002-9939-98-04203-8

Neither first countable nor Cech-complete spaces are maximal Tychonoff connected
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by D. B. Shakhmatov, M. G. Tkačenko, V. V. Tkachuk, S. Watson and R. G. Wilson PDF

Proc. Amer. Math. Soc. 126 (1998), 279-287 Request permission

Abstract:

A connected Tychonoff space $X$ is called maximal Tychonoff connected if there is no strictly finer Tychonoff connected topology on $X$. We show that if $X$ is a connected Tychonoff space and $X\in \{$locally separable spaces, locally Čech-complete spaces, first countable spaces$\}$, then $X$ is not maximal Tychonoff connected. This result is new even in the cases where $X$ is compact or metrizable.

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