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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 ...
[c] In any topological vector space (TVS), a compact subset is complete. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are not closed. If A and B are disjoint compact subsets of a Hausdorff space X, then there exist disjoint open sets U and V in X such that A ⊆ U and B ⊆ V. A continuous bijection from a ...
The σ-locally finite notion is a key ingredient in the Nagata–Smirnov metrization theorem, which states that a topological space is metrizable if and only if it is regular, Hausdorff, and has a σ-locally finite base. [9] [10] In a Lindelöf space, in particular in a second-countable space, every σ-locally finite collection of sets is ...
The reverse implications do not hold, for example, standard Euclidean space (R n) is σ-compact but not compact, [5] and the lower limit topology on the real line is Lindelöf but not σ-compact. [6] In fact, the countable complement topology on any uncountable set is Lindelöf but neither σ -compact nor locally compact. [ 7 ]
The product of a CG-2 space and a locally compact Hausdorff space is CG-2. [ 28 ] [ 29 ] When working in a category of compactly generated spaces (like all CG-1 spaces or all CG-2 spaces), the usual product topology on X × Y {\displaystyle X\times Y} is not compactly generated in general, so cannot serve as a categorical product .
Any compact group is locally compact.. In particular the circle group T of complex numbers of unit modulus under multiplication is compact, and therefore locally compact. The circle group historically served as the first topologically nontrivial group to also have the property of local compactness, and as such motivated the search for the more general theory, presented here.
In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space. [1] These kinds of fields were originally introduced in p-adic analysis since the fields Q p {\displaystyle \mathbb {Q} _{p}} are locally compact topological spaces constructed from the norm | ⋅ | p {\displaystyle |\cdot |_{p ...
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945. [1]