Locally finite – equivalent (?) definition

general-topology

A collection $(U_i)_i$ of open subsets of a topological space $X$ is called locally finite if every point $x \in X$ has a neighborhood $V_x$ that intersects only finitely many of the $U_i$'s. Clearly this implies that $x$ itself only lies in finitely many $U_i$'s, but the converse is not true in general, I think. Is there a good counterexample? And are there assumptions on $X$ under which the converse holds, i.e., local finiteness can be checked on points themselves, instead of on neighborhoods of points?

Best Answer

Let our collection consist of open intervals $(-1,1)$ and $\left(\dfrac1{n+1},\dfrac1n\right)$ for all $n\in\mathbb N$. Then clearly any point belongs to finitely many $U_i$, but good luck finding an open neighborhood of $0$ that intersects finitely many of them.

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