# Math Help - continuity in topological spaces

1. ## continuity in topological spaces

prove that if f:X->Y is continuous and if S is a subspace of X, then the restriction f|s: S->Y is continuous.

how would i go about doing this problem? can i create a mapping from S to X, then say S is open and use the fact that f is continuous?

2. Originally Posted by squarerootof2
prove that if f:X->Y is continuous and if S is a subspace of X, then the restriction f|s: S->Y is continuous.

how would i go about doing this problem? can i create a mapping from S to X, then say S is open and use the fact that f is continuous?
If U is an open subset of Y then its inverse image $f^{-1}(U)$ is open in X; and $\{x\in S:f(x)\in U\} = f^{-1}(U)\cap S$, which is relatively open in S. That's all there is to it!