Characterization of the function classes C(Y)|X

Abstract

Let X be a topological space and let Φ ⊂ C(X). Then there exists a topological space Y containing X as a subspace and such that Φ = C(Y)¦X, if and only if Φ is weakly composition closed, i.e., for any index set I, any fi ∈ Φ (i ∈ I) and any continuous map k : RI → Rwe have k ∘ 〈fi〉 ∈ Φ, where 〈fi〉 : X → RI is the map with i-th coordinate fi. The analogous statement is valid for functions to any T1 space, rather than to R, and even we can consider functions to any set of T1 spaces, and then a generalization of the above statement is valid, with a suitably defined weak composition closedness property. We also show that some earlier results on characterization of function classes Φ ⊂ C(X) of the form C(Y)¦X, with Y some extension of a given topological space X, and on the characterization of function classes C(〈X, T〉), with T some topology on a given set X, respectively, can be generalized in an analogous way as above, by means of composition properties analogous to the above one or by filter closedness (for functions to any set of T3 spaces, or to any set of topological spaces, respectively).