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A completely regular space
which is the $T_{1}$-complement of itself

Author: Stephen Watson
Journal: Proc. Amer. Math. Soc. 124 (1996), 1281-1284
MSC (1991): Primary 54A10, 05C20; Secondary 54B15, 54A25
MathSciNet review: 1343729
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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 [Enhancements On Off] (What's this?)

  • 1. B. A. Anderson and D. G. Stewart. $T_{1}$-Complements of $T_{1}$ Topologies. Proc. Amer. Math. Soc., 23:77--81, October 1969. MR 39:6240
  • 2. E. F. Steiner and A. K. Steiner. Topologies with $T_{1}$-complements. Fund. Math., 61:23--28, 1967. MR 37:5840

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Additional Information

Stephen Watson
Affiliation: Department of Mathematics and Statistics, York University, North York, Ontario, Canada M3J 1P3

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
Article copyright: © Copyright 1996 American Mathematical Society