Discrete topology is defined as: $P(X)=(A:A\subset X)$ Let X be an arbitrary non empty set, $T\subset P(X)$. It should also follow three axioms of union, intersection and $\phi$ and X should be in the $T$.
Q1) Does discrete topology has to have all sets as open?
Q2) I was trying to understand, how this discrete topology is the finest/strongest form of topology and all the rest of the topologies lie in between discrete and $\phi$. We can even define a topological space which includes open and closed sets, would not this be more finer than discrete topology.