Compactification (mathematics)

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For the concept of compactification in physics, see compactification (physics)

In mathematics, compactification is applied to topological spaces to make them compact spaces. The methods of compactification are various, but each is a way of controlling points from going off to infinity by in some way verifying a limit into a point or points, or preventing such an escape.

1 A simple, slightly handwavy example
2 Compactification in general topology
3 Compactification and discrete subgroups of Lie groups
4 Other compactification theories

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A simple, slightly handwavy example

We all know that the circle is a compact space in the plane. It is closed and bounded, which due to the special properties of the euclidean plane, means that is compact. What we would like to do is to show that we can adjust the line (which is not bounded) and identify it with the circle, making it a compact in a slightly modified topology which will preserve the topology everywhere except where we have modified it.

The construction is to add on a single point "infinity". Take a point on the circle and represent it by its degree value, in radians, going from -π to π for simplicity. Identify each point θ on the circle with the corresponding point on the real line tan(θ/2). This function is undefined at the point π, since tan(π/2) is undefined there; we will identify this point with our "infinity" point.

Note that since tangents and inverse tangents are both continuous, our identification function is a homeomorphism. It is not difficult to show that it remains a homeomorphism even when we consider our little tweak of adding "infinity". Topological properties, including compactness, are preserved over homeomorphisms, and since we know the circle is compact, we know that our modified line is compact, in fact, it is isomorphic to a circle.

Compactification, and in particular, one-point compactification can largely be considered a generalization of this process.

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Compactification in general topology

It is often useful to embed topological spaces in compact spaces, because of the strong properties compact spaces have. An embedding of a topological space X as a dense subset of a compact space is called a compactification of X.

Of particular interest are Hausdorff compactifications, i.e., compactifications in which the compact space is Hausdorff. A topological space has a Hausdorff compactification if and only if it is Tychonoff. Moreover, there is a unique (up to homeomorphism) "most general" compactification, the Stone-Čech compactification of X, denoted by βX. The space βX is characterized by the universal property that any continuous function from X to a compact Hausdorff space K can be extended to a continuous function from βX to K in a unique way. More explicitly, βX is a compact Hausdorff space for which the induced topology on X by βX is the same as the topology on X, and for any continuous map <math>f:X\to Y</math>, where Y is a compact Hausdorff space, there is a unique continuous map <math>g:\beta X\to Y</math> for which g restricted to X is identically f. The Stone-Čech compactification can be constructed explicitly as follows: let C be the set of continuous functions from X to [0,1]. 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. This is the Stone-Čech compactification.

For any non-compact space X the (Alexandroff) one-point compactification of X is obtained by adding an extra point ∞ (often called a point at infinity) and defining the open sets of the new space to be the open sets of X together with the sets of the form G U {∞}, where G is an open subset of X and X \ G is compact. The one-point compactification of X is Hausdorff if and only if X is Hausdorff and locally compact.

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Compactification and discrete subgroups of Lie groups

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. In n-dimensional Euclidean space the same questions can be posed, for example about GL_n(R)/GL_n(Z). This is harder to compactify. There is a general theory, the Borel-Serre compactification, that is now applied.

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