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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: 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. 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, since tan(π/2) is undefined; we will identify this point with our point ∞. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Compactification (mathematics)」の詳細全文を読む スポンサード リンク
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