Topology of a metric space

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 where two and . Then I need to show that there exists open sets

which statifies that 1) 2) and 3)

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

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 then d(x,y) > 0.

So the way to show 1) and possible 2) isn't that to claim that for set 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