[[Hausdorff-compact space]]
# The intersection of nested inhabited Hausdorff-compact sets is inhabited

Let $X$ be a [[Hausdorff space]] and $(K_i)_{i=1}^\infty$ be a strictly decreasing sequence of inhabited [[Compact space|compact]] subsets of $X$
(i.e. such that $\0 \neq K_{i} \supseteq K_{i+1}$ for all $i \in \mathbb{N}$).
Then #m/thm/topology 
$$
\begin{align*}
\bigcap_{i=1}^\infty K_{i} \neq \0
\end{align*}
$$

> [!check]- Proof
> Follows immediately from [[Compact space#Complement characterisation]].
> <span class="QED"/>

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