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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...