# continuity in topological spaces

• May 25th 2008, 10:17 PM
squarerootof2
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?
• May 26th 2008, 01:12 AM
Opalg
Quote:

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!