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
- Proc. Amer. Math. Soc. 128 (2000), 287-297
- DOI: https://doi.org/10.1090/S0002-9939-99-04984-9
- Published electronically: July 27, 1999
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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).References
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Bibliographic 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.
- Email: mich@xanum.uam.mx
- Vladimir V. Tkachuk
- Email: vova@xanum.uam.mx
- Richard G. Wilson
- Address at time of publication: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México 20, D.F.
- Email: rgw@xanum.uam.mx
- Ivan V. Yaschenko
- Affiliation: Moscow Center for Continuous Mathematical Education, B. Vlas’evskij, 11, 121002, Moscow, Russia
- Email: ivan@mccme.ru
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 287-297
- MSC (1991): Primary 54H11, 54C10, 22A05, 54D06; Secondary 54D25, 54C25
- DOI: https://doi.org/10.1090/S0002-9939-99-04984-9
- MathSciNet review: 1616605