In topology, a **locally compact space** is a topological space that, at each point, resembles compact spaces in some way. More formally, a topological space \( X \) is said to be locally compact if every point in \( X \) has a neighborhood that is compact. Here's a breakdown of the concept: 1. **Neighborhood**: A neighborhood of a point \( x \in X \) is any open set that contains \( x \).
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A space is locally compact when every point has a base of neighbourhoods containing compact neighbourhoods. A compact Hausdorff space is locally compact because regular separation produces .