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No submaximal topology on a countable set is |
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| Authors: | Mikhail G. Tkacenko Vladimir V. Tkachuk Richard G. Wilson Ivan V. Yaschenko |
| Affiliation: | Departamento de Matematicas, Universidad Autónoma Metropolitana, Av. Michoacan y La Purísima, Iztapalapa, A.P. 55-532,C.P. 09340, México D.F. ; Departamento de Matematicas, Universidad Autónoma Metropolitana, Av. Michoacan y La Purísima, Iztapalapa, A.P. 55-532, C.P. 09340, México D.F. ; Departamento de Matematicas, Universidad Autónoma Metropolitana, Av. Michoacan y La Purísima, Iztapalapa, A.P. 55-532,C.P. 09340, México D.F. ; Moscow Center for Continuous Mathematical Education, B. Vlas'evskij, 11, 121002, Moscow, Russia |
| Abstract: | Two  -topologies  and  given on the same set  , are called transversal if their union generates the discrete topology on  . The topologies  and  are  -complementary if they are transversal and their intersection is the cofinite topology on  . We establish that for any connected Tychonoff topology there exists a connected Tychonoff transversal one. Another result is that no  -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). |
| Keywords: | Transversal topology $ T_{1}$-complement connected space strongly $sigma $-discrete space |
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