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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).

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Quote from J. Väisälä's book "Lectures on n-dimensional quasiconformal mappings":

Let $D$ be a domain in $\overline{\mathbb R}^n$ and let $b\in \partial D$. [...] $D$ is finitely connected at $b$ if $b$ has arbitrarily small neighborhoods $U$ such that $U\cap D$ has a finite number of components.

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I think I found something suitable from Munkres's Topology; $X$ is "locally finite (path-)component" according to my definition if and only if the partition $X/\sim$ is locally finite, where $\sim$ is the (path-)connectivity equivalence relation.

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