On the injective C(T)-spaces

AbstractWe prove some topological properties of completely regular (esp. locally compact) topological spaces T such that the locally convex (esp. Fréchet) space C(T) equipped with the compact-open topology is injective (i.e., complemented in every locally convex space containing it isomorphically).Examples are presented which show that the results obtained here are optimal and that the present case differs essentially from the compact case studied by Amir. In particular, we give an example of a Fréchet injective space C(T) such that T contains no open non-empty extremally disconnected subset