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It is very useful to embed topological spaces in compact spaces, because of the strong properties compact spaces have. In general, the theory is limited to compactifications of Hausdorff spaces. For a locally compact space, there exists the one-point compatification: add a point at infinity, and define its neighborhoods as all co-compact sets. Because of local compactness, this topology is Haussdorf. Every Tychonoff space has the Stone-Cech compactifications, which is defined by the property that every continuous function to the reals can be continued to the compactification. Note that this does not hold for the one-point compactification: the one-point compactification of the Real line is a circle, but the function x |-> x cannot be continued continuously. Only Tychonoff spaces have Haussdorf compactifications, since a Hausdorff compact space is Tychonoff, and a subspace of a Tychonoff space is Tychnoff. This means that this is an alternative definition: a Tychonoff space is a Hausdorff space with 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 natural partial order on the equivalence classes of Hausdorff compactifications of any given Tychonoff space X, and it can be shown that under this partial order there is a unique maximal equivalence class. Any member of this equivalence class is called a Stone-Čech compatification of X, and is often denoted βX. βX has the characteristic property that that any continuous function from X to a compact Hausdorff space K can be extended to a continuous function from βX to K. This extension is unique. 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. |
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