A completely regular space which is the $T_1$-complement of itself
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- by Stephen Watson PDF
- Proc. Amer. Math. Soc. 124 (1996), 1281-1284 Request permission
Abstract:
Two topologies $\tau$ and $\sigma$ on a fixed set are $T_{1}$-complements if $\tau \cap \sigma$ is the cofinite topology and $\tau \cup \sigma$ is a sub-base for the discrete topology. In 1967, Steiner and Steiner showed that of any two $T_{1}$-complements on a countable set, at least one is not Hausdorff. In 1969, Anderson and Stewart asked whether a Hausdorff topology on an uncountable set can have a Hausdorff $T_{1}$-complement. We construct two homeomorphic completely regular $T_{1}$-complementary topologies.References
- B. A. Anderson and D. G. Stewart, $T_{1}$-complements of $T_{1}$ topologies, Proc. Amer. Math. Soc. 23 (1969), 77–81. MR 244927, DOI 10.1090/S0002-9939-1969-0244927-5
- E. F. Steiner and A. K. Steiner, Topologies with $T_{1}$-complements, Fund. Math. 61 (1967), 23–28. MR 230277, DOI 10.4064/fm-61-1-23-28
Additional Information
- Stephen Watson
- Affiliation: Department of Mathematics and Statistics, York University, North York, Ontario, Canada M3J 1P3
- Email: stephen.watson@mathstat.yorku.ca
- Received by editor(s): July 1, 1992
- Received by editor(s) in revised form: October 4, 1994
- Additional Notes: This work has been supported by the Natural Sciences and Engineering Research Council of Canada
- Communicated by: Franklin D. Tall
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1281-1284
- MSC (1991): Primary 54A10, 05C20; Secondary 54B15, 54A25
- DOI: https://doi.org/10.1090/S0002-9939-96-03524-1
- MathSciNet review: 1343729