Formal Definition
Let X be a topological space. Most commonly X is called locally compact, if every point of X has a compact neighbourhood.
There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular). But they are not equivalent in general:
- 1. every point of X has a compact neighbourhood.
- 2. every point of X has a closed compact neighbourhood.
- 2‘. every point of X has a relatively compact neighbourhood.
- 2‘‘. every point of X has a local base of relatively compact neighbourhoods.
- 3. every point of X has a local base of compact neighbourhoods.
Logical relations among the conditions:
- Conditions (2), (2‘), (2‘‘) are equivalent.
- Neither of conditions (2), (3) implies the other.
- Each condition implies (1).
- Compactness implies conditions (1) and (2), but not (3).
Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equivalent to it when X is Hausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact.
Authors such as Munkres and Kelley use the first definition. Willard uses the third. In Steen and Seebach, a space which satisfies (1) is said to be locally compact, while a space satisfying (2) is said to be strongly locally compact.
In almost all applications, locally compact spaces are also Hausdorff, and this article is thus primarily concerned with locally compact Hausdorff (LCH) spaces.
Read more about this topic: Locally Compact Space
Famous quotes containing the words formal and/or definition:
“Good gentlemen, look fresh and merrily.
Let not our looks put on our purposes,
But bear it as our Roman actors do,
With untired spirits and formal constancy.”
—William Shakespeare (15641616)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (18421910)