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

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.