In mathematics, compactification is the process or result of enlarging a topological space to make it compact. 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". For other meanings of mathematics or math, see mathematics (disambiguation). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...

A nonrigorous example

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. It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by ∞. The resulting compactification can be thought of as a circle (which is compact as a closed and bounded subset of the Euclidean plane). Every sequence that ran off to infinity in the real line will then converge to ∞ in this compactification. In mathematics, the real line is simply the set of real numbers. ...

Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (-π,π) on the x-axis; then bend the ends of this interval upwards (in positive y-direction) and move them towards each other, until you get a circle with one point (the topmost one) missing. This point is our new point ∞ "at infinity"; adding it in completes the compact circle. In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... This article is about the number. ...

A bit more formally: we represent a point on the unit circle by its angle, in radians, going from -π to π for simplicity. Identify each such point θ on the circle with the corresponding point on the real line tan(θ/2). This function is undefined at the point π/2, since tan(π/2) is undefined there; we will identify this point with our point ∞. An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ... The radian is a unit of plane angle. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...

Since tangents and inverse tangents are both continuous, our identification function is a homeomorphism between the real line and the unit circle without ∞. What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below. It is also possible to compactify the real line by adding two points, +∞ and -∞; this results in the extended real line. In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of...

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. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ...

Alexandroff one-point compactification

For any non-compact space X the (Alexandroff) one-point compactification of X is obtained by adding one 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 such that X G is compact. The one-point compactification of X is Hausdorff if and only if X is Hausdorff and locally compact. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ...

It is slightly surprising that every topological space can be compactified in this way using a single point at infinity, because one usually thinks of compact spaces as "small".

Stone-Čech compactification

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. In this case, there is a unique (up to homeomorphism) "most general" Hausdorff 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 containing X such that the induced topology on X by βX is the same as the given topology on X, and for any continuous map f:X → K, where K is a compact Hausdorff space, there is a unique continuous map g:βX → K for which g restricted to X is identically f. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... Wikipedia does not have an article with this exact name. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ...

The Stone-Čech compactification can be constructed explicitly as follows: let C be the set of continuous functions from X to the closed interval [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. In mathematics, Tychonoffs theorem states that the product of any collection of compact topological spaces is compact. ...

Projective space

Real projective space RPⁿ is a compactification of Euclidean space Rⁿ. For each possible "direction" in which points in Rⁿ 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 RP¹. Note however that the projective plane RP² is not the one-point compactification of the plane R² since more than one point is added. In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...

Complex projective space CPⁿ is also a compactification of Cⁿ; the Alexandroff one-point compactification of the plane C is (homeomorphic to) the complex projective line CP¹, which in turn can be identified with a sphere, the Riemann sphere. In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ... A rendering of 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 RP² intersect in precisely one point, a statement that is not true in R². Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

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. Look up discrete in Wiktionary, the free dictionary. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...

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'. In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as HΓ where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices. ... In common parlance, a cusp is an important moment usually regarded as a decision point upon which consequent events are determined. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... See lattice for other meanings of this term, both within and without mathematics. ...

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. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

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 a projective line with inversive ring geometry. Let X be a non-compact topological space. ... In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H. The concept... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ... In mathematics, inversive ring geometry is the extension, to the context of associative rings, of the concepts of Projective line, homogeneous coordinates, projective transformations, and Cross-ratio, concepts usually built upon rings that happen to be fields. ...