Definition of discrete topology

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.

• This raises a doubt for me. Can a closed set $A \subset X$ be a part of (X,$T$) discrete topology – vikram Sep 27 '14 at 15:34
• any set $A \subset X$ is part of the discrete topology. Hence any set $A$ is open. But also $X-A$ is a subset, hence also open. Therefore $A$ is as complement of its own open complement closed. – Daniel Valenzuela Sep 27 '14 at 15:37