Stone-Čech extension property

Every continuous map from a space to a compact Hausdorff space extends continuously to its Stone-Čech compactification. Uniqueness follows from density of the original space and the Hausdorff property of the target. Two Hausdorff compactifications with this property have comparison maps both ways; their composites fix the dense original space and therefore are identity maps, proving equivalence of the compactifications.