In particular, notice that each singleton set is open.
What is the difference between a hausdorf space and a discrete topological space?
I know any two points of a hausdorf space have disjoint neighbourhoods- would that not mean that the space be a discrete topology?
I have read that hausdorf can be made more fine but would that not mean the more coarse space could have elements that do not have disjoint neighbourhoods?
Every discrete space is hausdorff, but not every hausdorff space is discrete.
Take for example the set of real numbers with the natural topology, i.e. the topology which has all open intervals as a basis. This topology is Hausdorff, because for any two different points the open intervals and are disjoint open neighbourhoods of x resp. y.
But it is not discrete, because the singletons are not open.