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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Proposition one-point extension of locally compact space is Hausdorff precisely if original space is Let X X be a locally compact topological space.

Also, the image of a compact resp. The fact that large and interesting classes of non-compact spaces aledandroff in fact have compactifications of particular sorts makes compactification a common technique in topology. Every locally compact Hausdorff space is homemorphic to a open topological subspace of a compact topological space.

Alexandroff extension – Wikipedia

Akexandroff X X be any topological space. You are commenting using your WordPress. From Wikipedia, the free encyclopedia. August 12, at For a simple example: If X is a Hausdorff and locally compact space which is not compact, then the Alexandroff extension Y is the unique one-point compactification of X which is Hausdorff.

Regarding the first statement: Fompactification Alexandroff one-point compactification of R we constructed in the example above is in fact homeomorphic to RP 1.

For each possible “direction” in which points in R n can “escape”, one new point at infinity is added but each direction is identified with its opposite. Intuitively, the process can be pictured as follows: If is an open cover of X compactirication each U i is locally compact, then so is X. K-topologyDowker space.


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: Proposition inclusion into one-point extension is open embedding Alexxndroff X X be a topological space.

Our discussion above tells us: I try to motivate every definition I make, to the best of my ability. CW-complexes are paracompact Hausdorff spaces. The inclusion map c: Let be an open cover. Proof The unions and finite intersections of the open subsets inherited from X X poimt closed among themselves by the assumption that X X is a topological space.

Compactification (mathematics) – Wikipedia

The cusps stand in for those different ‘directions to infinity’. Example every locally compact Hausdorff space is an open subspace of a compact Hausdorff space Every locally compact Hausdorff space is homemorphic compactifiction a open topological subspace of a compact topological space.

In the study of discrete subgroups of Lie groupsthe quotient space of cosets is often a candidate for more subtle compactification to preserve structure at a richer level than just topological. Upon doing that, we immediately run into the problem of uniqueness, i. In this context and in view of the previous case, one usually writes. From Wikipedia, the free encyclopedia.


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Then the evident inclusion function i: This space is not compact; in a sense, points can go off to infinity to the left or to the right. The following alternative definition of local compactness is rather common.

This example already contains the key concepts of the general compctification. This is harder to compactify.

Of particular interest are Hausdorff compactifications, i. By using this site, you agree to the Terms of Use and Privacy Policy.

Let X be a Hausdorff space. Indeed, a union of sets of the first type is of the first type. Since the compacrification is compact by Tychonoff’s theoremthe closure of X as a subset of that space will also be compact.

For the converse, assume that X X is Hausdorff. It extends X by one point to give a compact space Y. The easiest way is to add just one point. Retrieved from ” https: Then the evident inclusion function. Thus a space can only admit a one-point compactification if it is locally compact, noncompact and Hausdorff. Hausdorff spaces are sober. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification.