TITLE: Why, in this case, if G Hausdorff then G is discrete?
QUESTION [0 upvotes]: Let $G$ a topological group. It is Hausdorff and locally compact. Why , if every point has a neighborhood that contains a finite number of points, then G is discrete?

REPLY [4 votes]: This does not depend on the group structure at all.
Claim. Let $X$ be a Hausdorff topological space such that every point has a finite neighborhood. Then the topology is discrete.
Proof. For every $x\in X$ let $U_x=\{x, u_1,\ldots,u_n\}$ be a finite open set containing $x$. Apply the Haudorff property to each of the pairs $(x,u_i)$ in turn to obtain open sets separating $x$ from each of the $u_i$. Intersect $U_x$ with the sets separating $x$ from the $u_i$ to obtain the singleton set $\{x\}$, which is open since it is a finite intersection of open sets.
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