Basis for the topology on ultrafilter α.

The basic open sets for the topology on ultrafilters are open.

The basic open sets for the topology on ultrafilters are also closed.

Every ultrafilter u on ultrafilter α converges to a unique point of ultrafilter α, namely mjoin u.

pure : α → ultrafilter α defines a dense inducing of α in ultrafilter α.

pure : α → ultrafilter α defines a dense embedding of α in ultrafilter α.

The extension of a function α → γ to a function ultrafilter α → γ. When γ is a compact Hausdorff space it will be continuous.

The value of ultrafilter.extend f on an ultrafilter b is the unique limit of the ultrafilter b.map f in γ.

The Stone-Čech compactification of a topological space.

The natural map from α to its Stone-Čech compactification.

The image of stone_cech_unit is dense. (But stone_cech_unit need not be an embedding, for example if α is not Hausdorff.)

The extension of a continuous function from α to a compact Hausdorff space γ to the Stone-Čech compactification of α.