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Compactification

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It is very 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.

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 Stone-Čech 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 non-compact space X has a one-point 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 one-point 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 Calabi-Yau spaces or on Orbifolds[?]



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