C-embedded subset ($C(Y)=C(X){|}_Y$)

A subset $Y\subseteq X$ is C-embedded if every continuous real-valued function on its subspace topology extends continuously to $X$. Closed subsets of normal spaces have this property by the Tietze extension theorem. The condition also makes sense for nonclosed subsets.

A subset is Cb-embedded if every bounded continuous real-valued function on it extends to a bounded continuous real-valued function on the ambient space. A bounded extension can be clipped to preserve the original bound. This is weaker than C-embeddedness: the natural numbers in their Stone-Čech compactification have the bounded extension property, but their unbounded identity function cannot extend to a compact space.