In mathematics , in general topology , compactification is the process or result of making a topological space into a compact 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". Consider the real line with its ordinary topology. This space is not compact; in a sense, points can go off to infinity to the left or to the right. The resulting compactification can be thought of as a circle which is compact as a closed and bounded subset of the Euclidean plane.

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In mathematics , in general topology , compactification is the process or result of making a topological space into a compact 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".

Consider the real line with its ordinary topology. This space is not compact; in a sense, points can go off to infinity to the left or to the right. The resulting compactification can be thought of as a circle which is compact as a closed and bounded subset of the Euclidean plane.

What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below. An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. It is often useful to embed topological spaces in compact spaces , because of the special properties compact spaces have. Embeddings into compact Hausdorff spaces may be of particular interest.

Since every compact Hausdorff space is a Tychonoff space , and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space. In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification. The fact that large and interesting classes of non-compact spaces do in fact have compactifications of particular sorts makes compactification a common technique in topology.

The one-point compactification of X is Hausdorff if and only if X is Hausdorff and locally compact. Of particular interest are Hausdorff compactifications, i. A topological space has a Hausdorff compactification if and only if it is Tychonoff. Then each point in X can be identified with an evaluation function on C. Thus X can be identified with a subset of [0,1] C , the space of all functions from C to [0,1].

Since the latter is compact by Tychonoff's theorem , the closure of X as a subset of that space will also be compact. Walter Benz and Isaak Yaglom have shown how stereographic projection onto a single-sheet hyperboloid can be used to provide a compactification for split complex numbers.

In fact, the hyperboloid is part of a quadric in real projective four-space. The method is similar to that used to provide a base manifold for group action of the conformal group of spacetime. Real projective space RP n is a compactification of Euclidean space R n.

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. The Alexandroff one-point compactification of R we constructed in the example above is in fact homeomorphic to RP 1.

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. Complex projective space CP n is also a compactification of C n ; the Alexandroff one-point compactification of the plane C is homeomorphic to the complex projective line CP 1 , which in turn can be identified with a sphere, the Riemann sphere.

Passing to projective space is a common tool in algebraic geometry because the added points at infinity lead to simpler formulations of many theorems. For example, any two different lines in RP 2 intersect in precisely one point, a statement that is not true in R 2. Compactification of moduli spaces generally require allowing certain degeneracies — for example, allowing certain singularities or reducible varieties.

This is notably used in the Deligne—Mumford compactification of the moduli space of algebraic curves. In the study of discrete subgroups of Lie groups , the quotient space of cosets is often a candidate for more subtle compactification to preserve structure at a richer level than just topological.

For example, modular curves are compactified by the addition of single points for each cusp , making them Riemann surfaces and so, since they are compact, algebraic curves.

Here the cusps are there for a good reason: the curves parametrize a space of lattices , and those lattices can degenerate 'go off to infinity' , often in a number of ways taking into account some auxiliary structure of level.

The cusps stand in for those different 'directions to infinity'. That is all for lattices in the plane. This is harder to compactify. There are a variety of compactifications, such as the Borel—Serre compactification , the reductive Borel-Serre compactification , and the Satake compactifications , that can be formed.

From Wikipedia, the free encyclopedia. Embedding a topological space into a compact space as a dense subset. Main article: One-point compactification. Topology 2nd ed. Prentice Hall. Annals of Mathematics. Relaxation in Optimization Theory and Variational Calculus. Berlin: W. Categories : Compactification. Hidden categories: Articles with short description. Namespaces Article Talk. Views Read Edit View history. Contribute Help Community portal Recent changes Upload file. By using this site, you agree to the Terms of Use and Privacy Policy.

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By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. It seems to me that one has to consider only proper continuous maps as morphisms.

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## Compactification (mathematics)

By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. The one-point compactification is unique when it exists; see the answer to this question. Sign up to join this community.

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## Alexandroff extension

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