Local Compactnesst In this section we study the notion of local compactness, and we prove the basic

Local Compactnesst In this section we study the notion of local compactness, and we prove the basic theorem that any locally compact Hausdorff space can be imbedded in a certain compact Hausdorff space, called its one-point compactification. Definition. A space X is said to be locally compact at x if there is some compact subset C of X that contains a neighborhood of x. If X is locally compact at each of its points, X is said simply to be locally compact. This section will be assumed in 35-3, in Chapter 7, and in 88-12. Chap 3 $ 3-8 ring of Local Compactness 183 Note that a compact space is automatically locally compact. EXAMPLE 1. The real line R is locally compact. The point x lies in some interval (a, b), which in turn is contained in the compact set (a, b). The sub- space of rational numbers is not locally compact, as you can check. Eon of EXAMPLE 2. The space R* is locally compact; the point x lies in some basis element (a,b) x.. x (a.. b.), which in turn lies in the compact set