On a certain class of complete regularity (Q1081150)

Given two topological spaces X and E, X is said to be E-completely regular if X is homeomorphic to a subspace of \(E^ m\) for some cardinal m. \({\mathfrak C}(E)\) represents the class of all E-complete regular spaces, and it is said to be a class of complete regularity. Let T be the space consisting of the points 0, 1, 2 where \(\{\) \(0\}\) and \(\{\) 1,2\(\}\) are the only proper open subsets. Theorem. \({\mathfrak C}(T)\) is the class of all zero-dimensional spaces.