No submaximal topology on a countable set is 𝐓_{1}-complementary

Tkacenko, Mikhail; Tkachuk, Vladimir; Wilson, Richard; Yaschenko, Ivan

doi:10.1090/S0002-9939-99-04984-9

No submaximal topology on a countable set is $\mathbf {T}_{\mathbf {1}}$-complementary
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by Mikhail G. Tkacenko, Vladimir V. Tkachuk, Richard G. Wilson and Ivan V. Yaschenko PDF

Proc. Amer. Math. Soc. 128 (2000), 287-297 Request permission

Abstract:

Two $T_{1}$-topologies $\tau$ and $\mu$ given on the same set $X$, are called transversal if their union generates the discrete topology on $X$. The topologies $\tau$ and $\mu$ are $T_{1}$-complementary if they are transversal and their intersection is the cofinite topology on $X$. We establish that for any connected Tychonoff topology there exists a connected Tychonoff transversal one. Another result is that no $T_{1}$-complementary topology exists for the maximal topology constructed by van Douwen on the rational numbers. This gives a negative answer to Problem 162 from Open Problems in Topology (1990).

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