This one-point compactification is also known as the Alexandroff compactification after a paper by Павел Сергеевич Александров (then. The one point compactification. Definition A compactification of a topological space X is a compact topological space Y containing X as a subspace. of topological spaces and the Alexandroff one point compactification. Some prop- erties of the locally compact spaces and one point compactification are proved.
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Recall that the stereographic projection S gives an explicit homeomorphism from the unit sphere minus the north pole 0,0,1 to the Euclidean plane. A topological space has a Hausdorff compactification if and only if it is Tychonoff.
Views Read Edit View history. In mathematicsin general topologycompactification is the process or result of making a topological space into a compact space. From Wikipedia, the free encyclopedia. Regarding the first point: Here the cusps are there for a good reason: In the more general situation, it may not really be a compactification and hence is called the one-point extension or Alexandroff extension.
For example, any two different lines in RP 2 intersect in precisely one point, a statement that is not true in R 2. Indeed, a union of sets of the first type is of the first type. This example already contains the key concepts of the general case.
As a pointed compact Hausdorff spacethe one-point compactification of X X may be described by a universal property:. It is often useful to embed topological spaces in compact spacesbecause of the special properties compact spaces have.
K-topologyAlexanfroff space. Compactification of moduli spaces generally require allowing certain degeneracies — for example, allowing certain singularities or reducible varieties.
This compactificwtion notably used in the Deligne—Mumford compactification of the moduli space of algebraic curves.
Since the latter is compact by Tychonoff’s theoremthe closure of X as a subset of that space will also be compact. Real projective space RP n is a compactification of Euclidean space R n.
The cusps stand in for those different ‘directions to infinity’.
one-point compactification in nLab
Retrieved from ” https: The product of infinitely many locally compact spaces is not locally compact in general to get a locally compact space, one uses the restricted product which leads to adeles and ideles in algebraic number theory. Regarding the second point: For example, a closed subset of a compact resp.
Lemma one-point extension is well-defined The topology on the one-point extension in def. Then the evident inclusion function. Let X X be a locally compact topological space. compactificaion
Continuity means maps closed subsets to closed subsets. Given a topological space X, we wish to construct a compact space Y by appending one point: The operation of one-point compactification is not a functor on the whole category of topological spaces. In this context and in view of the previous case, one usually writes. Passing to projective space is a common tool in algebraic geometry because the added points at infinity lead to simpler formulations of many theorems.