To prove that finitely generated spaces are corecompact, I used the following characterisation of finitely generated topological spaces:
In a finitely generated topological space every point has a smallest neighbourhood.
Now I was wondering how to prove this characterisation. The definition I use for finitely generated topological spaces is the following:
$\forall A \in \mathcal{P}(X), \forall x \in X: x \in cl(A) \Rightarrow \exists a \in A, x \in cl\{a\}$
I tried proving it by contradiction: Assume that there is no such thing as a smallest neighbourhood. Suppose that $\forall V \in \mathcal{V}(x) \exists W \in \mathcal{V}(x); W \subseteq V$. Can someone give me a hint so I can come to a contradiction? Or is there a nicer way to prove this?
As always, any hints or comments will be appreciated!