Continuity stuff

If is continuous and is a vector subspace of then is continuous.

How does one prove continuity with restricted into a set?

Let be continuous and then is continuous.

I need a hint for this one.
Thanks.

Hey Megus.

Can you show that the topologies have the same properties?

For example if you start with a simple region (and no holes of any kind), then the subset of any simple region is simple and the topologies are the same then continuity in the sub-space under the preservation of the topology will have the same properties of that as the non sub-space.

These problems are about relative topology. Given a topological space X, and a subset A, then the subspace topology for A is that a set V is open in A if and only if V = U intersect A for some U that's open in X.