Compactification (algebraic Geometry)
Compactification may refer to: * Compactification (mathematics), making a topological space compact * Compactification (physics), the "curling up" of extra dimensions in string theory See also * Compaction (other) Compaction may refer to: * Soil compaction, for mechanically induced compaction near the ground surface * Compaction of ceramic powders * Compaction (geology), part of the process of lithification involving mechanical dewatering of a sediment by ...

Compactification (mathematics)
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. A compact space is a space in which every open cover of the space contains a finite subcover. 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". An 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. Intuitively, the process can be pictured as follows ...
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Compactification (physics)
In theoretical physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic. Compactification plays an important part in thermal field theory where one compactifies time, in string theory where one compactifies the extra dimensions of the theory, and in two- or one-dimensional solid state physics, where one considers a system which is limited in one of the three usual spatial dimensions. At the limit where the size of the compact dimension goes to zero, no fields depend on this extra dimension, and the theory is dimensionally reduced. Compactification in quantum field theory Any two-dimensional scalar quantum field theory with a generic potential presents a universal feature, first unveiled by Campos Delgado and Dogaru, namely it is equivalent to a one-dimensional theory of partic ...
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