Stone–Čech compactification

An important problem in topology is how to enlarge a space by adding points so that certain kinds of limits exist. The Stone–Čech compactification of a space provides the most extensive such enlargement: it adds enough points to ensure the existence of all generalized limits, including those detected by nets or ultrafilters rather than ordinary sequences. The construction was implicitly introduced by Andrey Nikolayevich Tikhonov (1930) and explicitly described by Marshall Stone (1937) and Eduard Čech (1937).

In more detail, the Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space ${\displaystyle \beta X}$. The Stone–Čech compactification ${\displaystyle \beta X}$ is the unique "most general" compact Hausdorff space generated by X, in the sense that any continuous map from X into any other compact Hausdorff space factors uniquely through ${\displaystyle \beta X}$. If the space X is a Tychonoff space, the map from X to ${\displaystyle \beta X}$ is an embedding, and X can be identified as a dense subspace of ${\displaystyle \beta X}$. For a Tychonoff space, ${\displaystyle \beta X}$ is the largest compactification of X: any other Hausdorff compactification of X is a quotient space of ${\displaystyle \beta X}$. For infinite spaces, the structure of ${\displaystyle \beta X}$ is often extremely complex, and proving its existence generally requires the axiom of choice.