[[Neighbourhood basis]]
# 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 $S_{n} \sube S_{m}$ whenever $n \geq m$.

> [!info]+ Construction
> Let $(\tilde S_{n})_{n \in \mathbb{N}}$ a countable open [[neighbourhood basis]] of $a$.
> We define a new _nested_ open neighbourhood basis recursively so that $S_{1} = \tilde S_{1}$ and $S_{n + 1} = S_n \cap \tilde S_{n+1}$.


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