Let (X, τ) be a topolgical space. Let f: τ→R be a map that assigns real values to the elements of τ. Let (A,τ_A) be subspace of (X,τ). Let g:τ_A→R be another map that assigns real values to the element of subspace topology. My question is how the function g is related with function f given that f is a function on the parant topolgy and g is a function on the daughter topology. In the particular case in which I'm interested τ_A is a subset of τ.