Extension of maps and ordering

LetX be a topological space,Y a closed subspace and π:x→T, ψ:Y→T be two continuous maps. We shall say that ψ can be extended by π if there exists a continuous man η=ν(π, ψ):X→T such that: η| _x?y ?π, η| _Y =ψ. Clearly a similar definition can be given in the category of real or complex algebraic varietes. In this paper we give some sufficient conditions to ensure that map ψ can be extended by π. In particular we study the topological and the real algebraic case. It seems that the last setting is the more interesting.