A space is realcompact if and only if it can be embedded
homeomorphically as a closed subset in some (not necessarily finite)
Cartesian power of the reals, with the
product topology. Moreover, a (Hausdorff) 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 (Gillman, Jerison, p. 226).
For example
Lindelöf spaces are realcompact; in particular all subsets of are realcompact.
The (Hewitt) realcompactification υX of a topological space X consists of the real points of its
Stone–Čech compactification βX. A
topological spaceX is realcompact if and only if it coincides with its Hewitt realcompactification.
Write C(X) 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(Y) to C(X) 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(X).
In order that a
Hausdorff spaceX is compact it is necessary and sufficient that X is realcompact and pseudocompact (see Engelking, p. 153).
A space is realcompact if and only if it can be embedded
homeomorphically as a closed subset in some (not necessarily finite)
Cartesian power of the reals, with the
product topology. Moreover, a (Hausdorff) 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 (Gillman, Jerison, p. 226).
For example
Lindelöf spaces are realcompact; in particular all subsets of are realcompact.
The (Hewitt) realcompactification υX of a topological space X consists of the real points of its
Stone–Čech compactification βX. A
topological spaceX is realcompact if and only if it coincides with its Hewitt realcompactification.
Write C(X) 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(Y) to C(X) 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(X).
In order that a
Hausdorff spaceX is compact it is necessary and sufficient that X is realcompact and pseudocompact (see Engelking, p. 153).