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 StoneČech compatification 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.
Any noncompact space X has a onepoint compactification 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 open and X \ G is compact. The onepoint compactification of X is Hausdorff if and only if X is Hausdorff and locally compact.
In the String theory context, Compactification refers to "curling" up the extra dimensions ( six in the superstring theory) usually on CalabiYau spaces or on Orbifolds[?]
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