# Continuity stuff

• Oct 18th 2012, 04:41 PM
Megus
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.
• Oct 18th 2012, 10:57 PM
chiro
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 sub-space under the preservation of the topology will have the same properties of that as the non sub-space.
• Oct 18th 2012, 11:45 PM
johnsomeone
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.