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Proceedings of the American Mathematical Society

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



No submaximal topology on a countable set
is $\mathbf{T}_{\mathbf{1}}$-complementary

Authors: Mikhail G. Tkacenko, Vladimir V. Tkachuk, Richard G. Wilson and Ivan V. Yaschenko
Journal: Proc. Amer. Math. Soc. 128 (2000), 287-297
MSC (1991): Primary 54H11, 54C10, 22A05, 54D06; Secondary 54D25, 54C25
Published electronically: July 27, 1999
MathSciNet review: 1616605
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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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Additional Information

Mikhail G. Tkacenko
Affiliation: Departamento de Matematicas, Universidad Autónoma Metropolitana, Av. Michoacan y La Purísima, Iztapalapa, A.P. 55-532, C.P. 09340, México D.F.

Vladimir V. Tkachuk

Richard G. Wilson
Address at time of publication: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México 20, D.F.

Ivan V. Yaschenko
Affiliation: Moscow Center for Continuous Mathematical Education, B. Vlas’evskij, 11, 121002, Moscow, Russia

Keywords: Transversal topology, $ T_{1}$-complement, connected space, strongly $\sigma $-discrete space
Received by editor(s): January 15, 1998
Received by editor(s) in revised form: March 19, 1998
Published electronically: July 27, 1999
Additional Notes: This research was supported by Consejo Nacional de Ciencia y Tecnología (CONACYT) de México, grant 400200-5-3012PE.
Communicated by: Alan Dow
Article copyright: © Copyright 1999 American Mathematical Society