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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A one-point compactification of is given by the union of two circles which are tangent to each other.

A bit more formally: Let be an open cover. It follows that a noncompact Tychonoff space admits a minimal compactification if and only if it is locally compact.

Then the identity map f: Home About This Blog Contents. Note that a locally compact metric space is not necessarily complete, e.

We need to show that this has a finite subcover. Let X X be any topological space.

Compactification (mathematics)

The topology on the one-point extension in def. Let X X be a locally compact topological space. What we need to show is that every locally compact Hausdorff spaces arises this way. From Wikipedia, the free encyclopedia. In the mathematical field of topologythe Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact.


Topology: One-Point Compactification and Locally Compact Spaces | Mathematics and Such

I try to motivate every definition I make, to the best of my ability. Regarding the first statement: You are commenting using your Twitter account.

By using this site, you agree to the Terms of Use and Privacy Policy. Conpactification this case it is called the one-point compactification or Alexandroff compactification of X. Now using de Morgan duality we find. That is all for lattices in the plane. Regarding the second point: As a pointed compact Hausdorff spacethe one-point compactification of X X may be described by a universal property:.

This construction presents the J-homomorphism in stable homotopy theory and is encoded for instance in the definition of orthogonal spectra.

Alexandroff extension

Indeed, a union of sets of the first type is of the first type. This space is not compact; in a sense, points can go off to infinity to the left or to the right. Note alexandrofd that the projective plane RP 2 is not the one-point compactification of the plane R 2 since more than one point is added. But then which is open in the Alexandroff extension. For each possible “direction” in which points in R n can “escape”, one new point at alexanfroff is added but each direction is identified with its opposite.

For the converse, assume that X X is Hausdorff.

This page was last edited on 23 Octoberat A topological compactifjcation has a Hausdorff compactification if and only if it is Tychonoff.


Remark If X X is Hausdorffthen it is sufficient to speak of compact subsets in def. Since finite unions of closed subsets are closed, this is again an open subset of X X.

Proof The unions and finite intersections of the open subsets compactificagion from X X are closed among themselves by the assumption that X X is a topological space. It is often useful to embed topological spaces alexandroft compact spacesbecause of the special properties compact spaces have. But merely boundedness is not enough to ensure compactness, since any metric can be massaged to a bounded one without affecting the underlying topology.

This is notably used in the Deligne—Mumford compactification of the moduli space of algebraic curves.

one-point compactification in nLab

You are commenting using your WordPress. 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.

Thus a space can only admit a one-point compactification if it is locally compact, noncompact and Hausdorff. In this context and in view of the previous case, one usually writes. Views Read Edit View history.