TITLE: How can I define a topology on the empty set?
QUESTION [6 upvotes]: We know that, indiscrete topology is the smallest topology. It has $2$ elements (they are the empty set and whole set). Suppose the given set is the empty set, then how can I define a topology on that set? Is it possible?

REPLY [3 votes]: If the underlying $X$ of a topological space is even allowed to be empty under our system of definitions, then $\{ \emptyset \}$ is the only such topology. Note that this is the same as the indiscrete topology $\{ \emptyset,X \}$ since $X=\emptyset$. It is also the same as the discrete topology $\mathcal{P}(X)$.