The intersection of nested inhabited Hausdorff-compact sets is inhabited

Let $𝑋$ be a Hausdorff space and $(𝐾 𝑖) \infty 𝑖 = 1$ be a strictly decreasing sequence of inhabited compact subsets of $𝑋$ (i.e. such that $\emptyset \neq 𝐾 𝑖 \supseteq 𝐾 𝑖 + 1$ for all $𝑖 \in ℕ$ ). Then topology

\infty ⋂ 𝑖 = 1 𝐾 𝑖 \neq \emptyset

Proof

Follows immediately from Complement characterisation.

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The intersection of nested inhabited Hausdorff-compact sets is inhabited

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