Continuous functions and dense subset proof...?

Hi A A,

You have to show that i) every point in f(X) is either in f(S) or ii) is a limit point of f(S). In the following, assume that y is in f(X) and f(x) = y:

i) If x is in S, the f(x) is in f(S)

ii) Suppose x is a limit point of S. The idea is that since there is a sequence of points x_n in X that tend to x, the sequence of points f(x_n) tends to f(x) because of continuity.

Hope that helps!

Daniel

My expensive fellowd192e0c4ad64a9c35fe32972477e4cd8 typically an unlimited dimensional vector area does no longer have a organic topology on it. What topology are you pondering on C[d192e0c4ad64a9c35fe32972477e4cd8d192e0c4ad64a9c35fe32972477e4cd8d192e0c4ad64a9c35fe32972477e4cd8]? then again: what's the area between applications you're speaking about? :-(

Since S is dense in X, cl(S) = X, where cl() denotes closure. Since f is continuous, f(cl(S)) ⊂ cl(f(S)), i.e., f(X) ⊂ cl(f(S)). Hence f(S) is dense in f(X).