Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. Let X be a topological space. Most commonly X is called locally compact, if every point x of X has a compact neighbourhood, i.e., there exists an open set U and a compact set K, such that x ∈ U ⊆ K {displaystyle xin Usubseteq K} . There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular). But they are not equivalent in general: