Realcompactifications

From Stone’s characterization of βX (Theorem 6.2), it can be seen that if f is a real valued bounded continuous function on X, then f can be uniquely extended over βX. Consequently, any continuous function f:X → [0,1] has a unique continuous extension over βX. In fact, βX can be characterized by this property. To see this we need only establish that if each continuous function f:X → [0,1] has a unique continuous extension over βX, then any continuous function f:X → Y where Y is a compact Hausdorff space has a continuous extension over βX.