
Continuity stuff
If $\displaystyle f: (M_1,d_1)\to(M_2,d_2)$ is continuous and $\displaystyle N_1$ is a vector subspace of $\displaystyle M_1$ then $\displaystyle f\bigg_{N_1}(N_1,d_1)\to (M_2,d_2)$ is continuous.
How does one prove continuity with $\displaystyle f$ restricted into a set?
Let $\displaystyle f: (M_1,d_1)\to (M_2,d_2)$ be continuous and $\displaystyle f(M_1)\subseteq N_2\subseteq M_2$ then $\displaystyle f: (M_1,d_1)\to(N_2,d_2)$ is continuous.
I need a hint for this one.
Thanks.

Re: Continuity stuff
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 subspace under the preservation of the topology will have the same properties of that as the non subspace.

Re: Continuity stuff
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.