I need help in such a problem, suppose it's rather simple one, but I have no skill in a field.. If $p: E\rightarrow B$ - fibration with a fibre F, then for every locally compact space $X$ an induced mapping $p_*:Map(X,E)\rightarrow Map(X,B)$ is a fibration. Also: $\Omega p:\Omega E\rightarrow\Omega B$ is a fibration.
I'd like to use the following theorem (Borsuk's thm): $X$ - locally compact, $j:A\rightarrow X$ - Borsuk's pair, then $j^*:Map(X,Y)\rightarrow Map(A,Y)$ - cofibration.
and somehow use the duality of co- and fibrations. Is this a good way to prove it?
Thanks and take care