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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Let X X be a topological space.
Since every compact Hausdorff space is a Tychonoff spaceand every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space.
Notify me of new comments via email. The inclusion map c: Definition one-point extension Let X X be any topological space. Compactfiication X X is Hausdorffthen it is sufficient to speak of compact subsets in def.
Let X X be any topological space. Extra stuff, structure, properties. The following alternative definition of local compactness is rather common.
Then the evident inclusion function i: Thus X can be identified with a subset of [0,1] Cthe space of all functions from C to [0,1]. Examples Any discrete topological space is locally compact and Hausdorff. The Alexandroff extension can be viewed as a functor from the category of topological spaces with proper continuous maps as morphisms to the category whose objects are continuous maps c: Also, the image of a compact resp.
But it does extend to a functor on topological spaces with proper maps between them.
This page was last edited on 8 Septemberat Consider the real line with its ordinary topology. Since finite unions of closed subsets are closed, this is again an open subset of X X.
Compactification (mathematics) – Wikipedia
Let be an open cover. A one-point compactification of is given by the union of two circles which are tangent to each other. If X is locally compact, then so is any open subset U. Proposition inclusion into one-point extension is open embedding Let X X be a topological space. The methods of compactification are various, but each is a way of controlling points from “going off to infinity” by in some way adding “points at infinity” or preventing such an “escape”.
Then c X is open in a compact Hausdorff space so is locally compact Hausdorff, hence its homeomorphic preimage X is also locally compact Hausdorff. A one-point compactification of [0, 1 is given by [0, 1]. 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.
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 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. Let X X be a locally compact topological space.
Then the identity map f: Relaxation in Optimization Theory and Variational Calculus. What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below. The cusps stand in for those different ‘directions to infinity’.
Alexandroff extension – Wikipedia
Regarding the third point: Upon doing that, we immediately run into the problem of uniqueness, i. Home About This Blog Contents. Checking that this gives us a topology.
Since the latter is compact by Tychonoff’s theoremthe closure of X as a subset of that space will also be compact. 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:. In particular, the Alexandroff extension c: An embedding of a loint space X as a dense subset of a compact space is called a compactification of X.
I try to motivate every definition I make, to the best of my ability. A bit more formally: Views Read Edit View history. For example, any two different lines in RP 2 intersect in precisely one point, a statement compactificagion is not true in R 2.