Hi, say X is a topological space with subspaces Y,Z , so that Y and Z are homotopic in X. Does it follow that there is a continuous map f:X→X with f(Y)=Z ? Do we need isotopy to guarantee the existence of a _homeomorphism_ h: X→X , taking Y to Z ? It seems like the chain of maps parameterized by t , for each x, in H: XxI→X with H(x,0)=Y and H(x,1)=Z would give us a path H(x,t) of continuous maps parameterized by t , by composing the maps at each stage t ; from t=0 to t=1 ; we get an infinite chain of maps, so that and that the composition H(x,s); s in [0,1] is a continuous map from X to X taking Y to Z. What if Y,Z are just curves in X ?