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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Extra stuff, structure, properties. Kolmogorov spaceHausdorff spaceregular spacenormal space. Cantor spaceMandelbrot space. K-topologyDowker space.
Warsaw circleHawaiian earring space. Hausdorff spaces are sober.
CW-complexes are paracompact Hausdorff spaces. Brouwer’s fixed point theorem. The one-point compactification is usually applied to a non- compact locally compact Hausdorff space. In the more general situation, it may not really be a compactification and hence is called the one-point extension or Alexandroff extension. Let X X be any topological space. If X X is Hausdorffthen it is sufficient to speak of compact subsets in def.
The topology on the one-point extension in def. The unions and finite intersections of the open subsets inherited from X X are closed among themselves by the assumption that X X is a topological space. Regarding the first statement: Under de Morgan duality. Regarding the second statement: Now using de Morgan duality we find. Since finite unions of closed subsets are closed, this is again an open subset of X Poinnt.
This follows because subsets are closed in a closed subspace precisely if they are closed in the ambient space and because closed subsets of compact spaces are compact. We need to show that this has a finite subcover. Let X X be a locally compact topological space. For the converse, assume that X X is Hausdorff. Let X X be a topological space. Then the evident inclusion function. Regarding the first point: Regarding the second point: Regarding the third point: Compactidication need to show that i: As a pointed compact Hausdorff spacethe one-point compactification of X X may be described by a universal property:.
The operation of one-point compactification is not a functor on the whole category of topological spaces. But it does extend to a functor on topological spaces with proper maps between them. By stereographic projection we have a homeomorphism.
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This construction presents the J-homomorphism in stable homotopy theory and is encoded for instance poknt the definition of orthogonal spectra. In this context and in view of the previous case, one usually writes. For a simple example: Every locally compact Hausdorff space is homemorphic to a alexandrofr topological subspace of a compact topological space. In one direction the statement is that open subspaces of compact Hausdorff spaces are locally compact see there for the proof.
Compactification (mathematics) – Wikipedia
What we need to show is that every locally compact Hausdorff spaces arises this way. So let X X be a locally compact Hausdorff space. Definition one-point extension Let X X be any topological space. Remark If X X is Hausdorffthen it is sufficient to speak of compact subsets in def.
Lemma one-point extension is well-defined The topology on the one-point extension in def.
Proof The unions on finite intersections of the open subsets inherited from X X are closed among themselves by the assumption that X X is a topological space. Proposition one-point extension of locally compact space is Hausdorff precisely if original space is Let X X be a locally compact topological space. Proposition inclusion into one-point extension is open embedding Let X X be a topological space.
Then the evident inclusion function i: Proof Regarding the first point: Example every locally compact Hausdorff space is an open subspace of a compact Hausdorff space Every locally compact Hausdorff space is homemorphic to a open topological subspace of a compact topological space. Proof In one direction the statement is that open subspaces of compact Hausdorff spaces are locally compact see there for the proof.