TITLE: A property of an open cover
QUESTION [7 upvotes]: Let $\{A_i\}_{i\in S}$ be an open cover for a regular topological space $X$.
 The family $\{A_i\}_{i\in S}$ is called locally finite if for every point $x\in X$ there exists a neighborhood $U$ of $x$ such that the set
$\{s\in S : U \cap A_s\neq \emptyset\}$ is finite.
Let the family $\{A_i\}_{i\in S}$ have the property that every point of $X$ is contained in only finitely many of $A_i'$s. Can we deduce that the family $\{A_i\}_{i\in S}$ is locally finite?

REPLY [7 votes]: In other words, you're asking if a point-finite open cover is locally finite. For a simple counterexample, take $X=\mathbb R,$ and the open cover $\mathcal A=\{\mathbb R,(\frac12,1),(\frac13,\frac12),(\frac14,\frac13),\dots\}.$