Definition:Topological Space

Let $$X$$ be any set and let $$\vartheta$$ be a topology on $$X$$.

Then $$\left({X, \vartheta}\right)$$ is called a topological space.

Notation
Some authors use the suboptimal $$\left\{{X, \vartheta}\right\}$$, which leaves it conceptually unclear as to which is the set and which the topology. This adds unnecessary complexity to the underlying axiomatic justification for the existence of the very object that is being defined.