Hi I have started a new subject this week called Topology, and it a bit harder than the stuff that I am used too, but anyway

We have metric space called $\displaystyle (M, d)$ where two $\displaystyle m_1, m_2 \in M$ and $\displaystyle m_1 \neq m_2$. Then I need to show that there exists open sets $\displaystyle S_1, S_2 \in \mathcal{T}_d$

which statifies that 1) $\displaystyle m_1 \in S_1$ 2) $\displaystyle m_2 \in S_2$ and 3) $\displaystyle S_1 \cap S_2 = \emptyset$

Firstly I know that the definition of open set or ball is as follows

$\displaystyle \{\forall x \in U \exists r > 0 : B_r(x) \subseteq U\}$ In other words: A set U is considered open if there for every element in the set is the center of an open ball of the set.

I also know from the metric subject field that if x,y are point in the set T and if they a$\displaystyle x \neq y$ then d(x,y) > 0.

So the way to show 1) and possible 2) isn't that to claim that for set $\displaystyle S_i$ to be open I must show that every point m_j on that set will be the center of an open ball?

I can see all the definitions that I need in my head I just need some assistance to connect them

Cheers

Julie