No submaximal topology on a countable set

is -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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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).

**[An]**B. A. Anderson,*A class of topologies with 𝑇₁-complements*, Fund. Math.**69**(1970), 267–277. MR**0281140****[AnSt]**B. A. Anderson and D. G. Stewart,*𝑇₁-complements of 𝑇₁ topologies*, Proc. Amer. Math. Soc.**23**(1969), 77–81. MR**0244927**, 10.1090/S0002-9939-1969-0244927-5**[Ar]**A. V. Arkhangel′skiĭ,*Continuous mappings, factorization theorems and spaces of functions*, Trudy Moskov. Mat. Obshch.**47**(1984), 3–21, 246 (Russian). MR**774944****[ArCo]**A. V. Arhangel′skiĭ and P. J. Collins,*On submaximal spaces*, Topology Appl.**64**(1995), no. 3, 219–241. MR**1342519**, 10.1016/0166-8641(94)00093-I**[vD]**E.K.van Douwen, Applications of maximal topologies, Topology and Its Applications, 1993, vol. 51, 125-139.**[En]**Ryszard Engelking,*General topology*, PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. MR**0500780****[St]**A. K. Steiner,*Complementation in the lattice of 𝑇₁-topologies*, Proc. Amer. Math. Soc.**17**(1966), 884–886. MR**0193033**, 10.1090/S0002-9939-1966-0193033-4**[StSt1]**E. F. Steiner and A. K. Steiner,*Topologies with 𝑇₁-complements*, Fund. Math.**61**(1967), 23–28. MR**0230277****[StSt2]**A. K. Steiner and E. F. Steiner,*A 𝑇₁-complement for the reals*, Proc. Amer. Math. Soc.**19**(1968), 177–179. MR**0231342**, 10.1090/S0002-9939-1968-0231342-2**[Wa1]**S. Watson, Problems I wish I could solve, Open Problems in Topology, ed. by J. van Mill and G.M. Reed, Elsevier Science Publishers B.V. (North Holland), 1990, 37-76. CMP**91:03****[Wa2]**Stephen Watson,*The number of complements in the lattice of topologies on a fixed set*, Topology Appl.**55**(1994), no. 2, 101–125. MR**1256214**, 10.1016/0166-8641(94)90112-0

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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.

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

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-04984-9

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