If $X$ is a metric space and $d$ is the metric s.t $d: X\rightarrow \mathbb R$ then a finite sum like $\sum_{k=1}^n d(a,x_k)$ for a fixed $a\in X$ is a valid equation. But we know that this kind of summation is not defined in an arbitrary non-metrizable topology. Disjoint union in topology comes closest to summation.So can I write something like $$\bigcup_{k=1}^nU(a,x_k) $$ where $U(a,x_k)$ is a nbd of $a$ that contains $x_k$ too.But such an union can be indefinitely large, the whole space infact but that I do not want; I want this union of nbd's as small as possible. How do I ensure that?
Thank you.