| Local Compactness |
Article Index for Local |
Website Links For Compact |
Information AboutLocal Compactness |
| CATEGORIES ABOUT LOCALLY COMPACT SPACE | |
| general topology | |
| properties of topological spaces | |
|
To be precise, a topological space ''X'' is locally compact Iff every Point has a Local Base of compact Neighborhood s, i.e. if every neighborhood of every point x of X contains a compact neighborhood of x. (Note that these neighborhoods do not have to be Open themselves but need only contain an open set containing the given point.) Other definitions may be found in the literature, as discussed in the section Non-Hausdorff Spaces below; however, this is the definition used in Wikipedia. The various definitions of local compactness all coincide for Hausdorff Space s. Almost all locally compact spaces studied in applications are Hausdorff, and this article is thus primarily concerned with locally compact Hausdorff spaces. EXAMPLES AND NONEXAMPLES Compact Hausdorff spaces Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article Compact Space . Here we mention only:
Locally compact Hausdorff spaces that are not compact The Euclidean Space s Rn (and in particular the Real Line R) are locally compact as a consequence of the Heine-Borel Theorem . Topological Manifold s share the local properties of Euclidean spaces and are therefore also all locally compact. This even includes Nonparacompact manifolds such as the Long Line . All Discrete Space s are locally compact and Hausdorff (they are just the Zero -dimensional manifolds). All Open or Closed Subset s of a locally compact Hausdorff space are locally compact in the Subspace topology. This provides several examples of locally compact subsets of Euclidean spaces, such as the Unit Disc (either the open or closed version). The space Q''p'' of ''p''-adic Numbers is locally compact for any Prime Number ''p'', because it is Homeomorphic to the Cantor Set minus one point. Thus locally compact spaces are as useful in ''p''-adic Analysis as in classical Analysis . Hausdorff spaces that are not locally compact As mentioned in the following section, no Hausdorff space can possibly be locally compact if it isn't also a Tychonoff Space ; there are some examples of Hausdorff spaces that aren't Tychonoff spaces in that article. But there are also examples of Tychonoff spaces that fail to be locally compact, such as:
The first two examples show that a subset of a locally compact space need not be locally compact, which contrasts with the open and closed subsets in the previous section. The last example contrasts with the Euclidean spaces in the previous section; to be more specific, a Hausdorff topological vector space is locally compact if and only if it is finite-dimensional (in which case it is a Euclidean space). This example also contrasts with the Hilbert cube as an example of a compact space; there is no contradiction because the cube cannot be a neighbourhood of any point in Hilbert space. FACTS ABOUT LOCALLY COMPACT HAUSDORFF SPACES As mentioned in the previous section, any Compact Hausdorff space is also locally compact, and any locally compact Hausdorff space is in fact a Tychonoff Space . Every locally compact Hausdorff space is a Baire Space . That is, the conclusion of the of every Union of Countably Many Nowhere Dense Subset s is Empty . A Subspace ''X'' of a locally compact Hausdorff space ''Y'' is locally compact If And Only If ''X'' can be written as the Set-theoretic Difference of two Closed Subset s of ''Y''. As a corollary, a Dense subspace ''X'' of a compact Hausdorff space ''Y'' is locally compact if and only if ''X'' is an Open Subset of ''Y''. Furthermore, if a subspace ''X'' of ''any'' Hausdorff space ''Y'' is locally compact, then ''X'' still must be the difference of two closed subsets of ''Y'', although the Converse needn't hold in this case. Quotient Space s of locally compact Hausdorff spaces are Compactly Generated . Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space. The point at infinity Since every locally compact Hausdorff space ''X'' is Tychonoff, it can be Embedded in a compact Hausdorff space b(''X'') using the Stone-Čech Compactification . But in fact, there is a simpler method available in the locally compact case; the One-point Compactification will embed ''X'' in a compact Hausdorff space a(''X'') with just one extra point. (The one-point compactification can be applied to other spaces, but a(''X'') will be Hausdorff If And Only If ''X'' is locally compact and Hausdorff.) The locally compact Hausdorff spaces can thus be characterised as the Open Subset s of compact Hausdorff spaces. Intuitively, the extra point in a(''X'') can be thought of as a point at infinity. The point at infinity should be thought of as lying outside every compact subset of ''X''. Many intuitive notions about tendency towards infinity can be formulated in locally compact Hausdorff spaces using this idea. |
|
|