Editing Compactification (mathematics) (section)

== 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 is [[homeomorphism|homeomorphic]] to a circle in the plane (which, as a closed and bounded subset of the Euclidean plane, is compact). Every sequence that ran off to infinity in the real line will then converge to ∞ in this compactification.  The direction in which a number approaches infinity on the number line (either in the - direction or + direction) is still preserved on the circle; for if a number approaches towards infinity from the - direction on the number line, then the corresponding point on the circle can approach ∞ from the left for example. Then if a number approaches infinity from the + direction on the number line, then the corresponding point on the circle can approach ∞ from the right.

Intuitively, the process can be pictured as follows: first shrink the real line to the [[open interval]] {{nowrap|(−[[pi|{{pi}}]], {{pi}})}} 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 [[radian]]s, going from −{{pi}} to {{pi}} for simplicity.  Identify each such point ''θ'' on the circle with the corresponding point on the real line [[tangent|tan]](''θ''/2).  This function is undefined at the point {{pi}}, since tan({{pi}}/2) is undefined; we will identify this point with our point&nbsp;∞.

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]].