We prove that, for every countable ordinal α ≥ 3, there exists countable completely regular spaces Xα and Yα such that the spaces Cp (Xα ) and Cp (Yα ) are borelian of class exactly Mα , but are not homeomorphic.
We give an example of a compact space X whose iterated continuous function spaces , are Lindelöf, but X is not a Corson compactum. This solves a problem of Gul’ko (Problem 1052 in [11]). We also provide a theorem concerning the Lindelöf property in the function spaces on compact scattered spaces with the th derived set empty, improving some earlier results of Pol [12] in this direction.
Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation with and a partition of unity subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction attains its extrema at vertices of P. Finally, a class of extremal functions on the metric space...