### A compact Hausdorff topology that is a T₁-complement of itself

Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X,{\tau}_{X})$ and $(Y,{\tau}_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that ${\tau}_{X}$ and ${f}^{-1}\left(U\right):U\in {\tau}_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size ${2}^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff...