Stone–Čech compactification

Summary

The Stone–Čech compactification $\beta X$ of a Tychonoff space $X$ is the largest Hausdorff compactification of $X$. It is characterised by a universal property. Every continuous map from $X$ to a compact Hausdorff space extends uniquely to $\beta X$. It links topology with algebra and set theory, for example through the description of $\beta \N$ by ultrafilters.