Nested neighbourhood basis

A nested (open) neighbourhood basis is a useful construction in many proofs involving the First countability axiom since whenever the axiom holds it is possible to construct such a neighbourhood basis and it has the property that $𝑆 𝑛 \subseteq 𝑆 𝑚$ whenever $𝑛 \geq 𝑚$ .

Construction

Let $(˜ 𝑆 𝑛) 𝑛 \in ℕ$ a countable open neighbourhood basis of $𝑎$ . We define a new nested open neighbourhood basis recursively so that $𝑆 1 = ˜ 𝑆 1$ and $𝑆 𝑛 + 1 = 𝑆 𝑛 \cap ˜ 𝑆 𝑛 + 1$ .

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Nested neighbourhood basis

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