Given a space, is there a notion of "how many" open sets contain a given point?

general-topology measure-theory

Sure, why not? You could let $O_x$ be the cardinality of the set of open sets containing x. I'm thinking this this cardinality should be the same for corresponding points h(x) and x under a homeomorphism h. That is $O_x=O_{h (x)}. $ So sort of a point wise topological invariant. ..But making this a global invariant (or what's normally meant by an invariant) appears impossible... because different points x and y will in general have different $O_x$ and $O_y$...

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