The intersection of inhabited nested sets may be empty

Consider a strictly decreasing sequence $(𝑆 𝑛) \infty 𝑛 = 1$ of inhabited subsets (i.e. such that $\emptyset \neq 𝑆 𝑛 ⊋ 𝑆 𝑛 + 1$ for $𝑛 \in ℕ$ ). Then it is possible that naïve

\infty ⋂ 𝑛 = 1 𝑆 𝑛 = \emptyset

Proof

For example, let
$𝑆 𝑛 = {𝑚 \in ℕ : 𝑚 \geq 𝑛} \subseteq ℕ$
for $𝑛 \in ℕ$ . Then clearly $(𝑆 𝑛) \infty 𝑛 = 1$ is strictly decreasing, but every $𝑚 \in ℕ$ has $𝑚 \notin 𝑆 𝑚 + 1$ . Therefore $⋂ \infty 𝑛 = 1 𝑆 𝑛 = \emptyset$ .

However, The intersection of nested inhabited Hausdorff-compact sets is inhabited.

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The intersection of inhabited nested sets may be empty

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