Realcompact space

Properties

• A space is realcompact if and only if it can be embedded homeomorphically as a closed subset in some Cartesian power of the reals, with the product topology. Moreover, a space is realcompact if and only if it has the uniform topology and is complete for the uniform structure generated by the continuous real-valued functions.
• For example Lindelöf spaces are realcompact; in particular all subsets of are realcompact.
• The realcompactification υX of a topological space X consists of the real points of its Stone–Čech compactification βX. A topological space X is realcompact if and only if it coincides with its Hewitt realcompactification.
• Write C for the ring of continuous real-valued functions on a topological space X. If Y is a real compact space, then ring homomorphisms from C to C correspond to continuous maps from X to Y. In particular the category of realcompact spaces is dual to the category of rings of the form C.
• In order that a Hausdorff space X is compact it is necessary and sufficient that X is realcompact and pseudocompact.