Does there exist in literature the notion of a "locally finite-component space"? That is, some topological space $X$ such that for all $x \in X$, there exists some open neighborhood $U$ of $x$ such that $U$ has a finite number of connected components? What about a finite number of path components?
The closest thing I've seen so far is that $X$ is locally path-connected if and only if every open subset of $X$ has open connected components (which are also precisely the path-connected components).