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. There is no unique way to do this.

The methods of compactification are various, but each is a way of controlling points from going off to infinity by in some way reifying a limit into a point or points, or preventing such an escape.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "":): {\displaystyle f:X\to Y} , where Y is a compact Hausdorff space, there is a unique continuous map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "":): {\displaystyle g:\beta X\to Y} 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 on C. 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.

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 GLn(R)/GLn(Z). This is harder to compactify. There is a general theory, the Borel-Serre compactification, that is now applied.

Other compactification theories

These include the theories of ends of a space and prime ends. Also some 'boundary' theories such as the collaring of an open manifold, Martin boundary, Silov boundary and Furstenberg boundary. The Bohr compactification of a topological group arises from the consideration of almost periodic functions. One can compactify a topological ring by forming its projective line with inversive ring geometry. de:Kompaktifizierung he:קומפקטיפיקציה