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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What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below. Under de Morgan duality. Let X X be a locally compact topological space. Note that a locally compact metric space is not necessarily complete, e.

Compactification (mathematics)

A bit more formally: Relaxation in Optimization Theory and Variational Calculus. For the converse, assume that X X is Hausdorff. Let X be a Hausdorff space. com;actification

If X is locally compact, then so is any open subset U. Note however that the projective plane RP 2 is not the one-point compactification of the plane R 2 since more than one point is added. Proposition inclusion into one-point extension is open embedding Let X X be a topological space.


Proof Regarding the first point: By using this site, you agree to the Terms of Use and Privacy Policy. Definition one-point extension Let Compactificatoin X be any 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.

Warsaw circleHawaiian earring space.

Alexandroff extension

Upon doing that, we immediately run into the problem of uniqueness, i. 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”.

Brouwer’s fixed point theorem. Then each point in X can be identified with an evaluation function on C. K-topologyDowker space.

Embeddings into compact Hausdorff spaces may be of particular interest. Moreover, if X were compact then c X would be closed in Y and hence not dense. This example already contains the key concepts of the general case.

The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. As a pointed compact Hausdorff spacethe one-point compactification compactificatoin X X may be described by a universal property:.

Thank you a lot for posting this. In the more general situation, it may not really be a compactification and hence is called the one-point extension or Alexandroff extension. 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.


A one-point compactification of is given by the union of two circles which are tangent to each other. Of particular interest are Hausdorff compactifications, i. If X is a Hausdorff and locally compact space which is not compact, then the Alexandroff extension Y is the pointt one-point compactification of X which is Hausdorff.

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Compactification (mathematics) – Wikipedia

Hausdorff spaces are sober. In particular, a disjoint union of locally compact spaces is locally compact.

A topological space has a Hausdorff compactification if and only if it is Tychonoff.

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. August 12, at