A locally compact group is a topological group whose underlying space is a locally compact space. In Haar measure and Fourier transform theory one normally also assumes it is a Hausdorff space. Local compactness supplies compact neighbourhoods, not just compact sets which algebraically generate the group.
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A **locally compact group** is a type of topological group that has the property of local compactness in addition to the group structure. Let's break down the definitions: 1. **Topological Group**: A group \( G \) is equipped with a topology such that both the group operation (multiplication) and the inverse operation are continuous.