Section 29: Local Compactness

1. A locally compact at a point space is a space that contains a compact subspace containing a neighborhood of the point.
2. A locally compact space is a space that is locally compact at each of its points.
3. A Hausdorff space is locally compact iff any neighborhood of any point contains a compact closure of a neighborhood of the point.

Two of the most well-behaved classes of spaces to deal with in mathematics are the metrizable spaces and the compact Hausdorff spaces... If a given space is not of one of these types, the next best thing one can hope for is that it is a subspace of one of these spaces... Thus arises the question: Under what conditions is a space homeomorphic with a subspace of a compact Hausdorff space?

1. One-point compactification of a locally compact Hausdorff space:
2. $X$ is locally compact Hausdorff iff $X$ is a subspace of a compact Hausdorff space $X\cup\{\infty\}$ .
3. If $X$ is locally compact Hausdorff then the one-point compactification is unique up to a homeomorphism that preserves points of $X$ .
4. One-point compactification seems to work for any Hausdorff space but is Hausdorff for locally compact spaces only.
5. Subspaces and local compactness:
6. A closed subspace of a locally compact space is locally compact.
7. An open subspace of a locally compact Hausdorff space is locally compact (and Hausdorff).
8. $X$ is locally compact Hausdorff iff it is homeomorphic to an open subspace of a compact Hausdorff space.
9. Continuous functions and local compactness:
10. The open continuous image of a locally compact space is locally compact.
11. In general, the continuous image of a locally compact space does NOT have to be locally compact.
12. A homeomorphism of two locally compact Hausdorff spaces can be extended to a homeomorphism of their one-point compactifications.
13. Quotient maps and local compactness:
14. If $p:X\to Y$ is a quotient map and $Z$ is locally compact and Hausdorff then $p\times\iota_{Z}$ is a quotient map.
15. If $p:A\to B,q:C\to D$ are quotient maps and $B,C$ (or $A,D$ ) are locally compact and Hausdorff then $p\times q$ is a quotient map.
16. Products and local compactness:
17. The finite product of locally compact spaces is locally compact.
18. The product of any family of spaces is locally compact $\Leftrightarrow$ all but finitely many of them are compact and those which are not compact are locally compact.
19. Topological groups and local compactness:
20. If a topological group is locally compact then any its quotient group is locally compact as well.