Definition

Let $X$ be a set and $\tau$ be a set of subsets of $X$. Then $\tau$ is called a topology on $X$, whose elements are called open sets, if it is subject to the following axioms:

(1) $X,\varnothing$ are open sets;

(2) Any finite intersection of open sets is open;

(3) Arbitrary unions of open sets are open.

A topological space is a pair $(X,\tau)$ consisting of a set $X$ and a topology $\tau$ on $X$. By abuse of notation, we will often simply denote by $X$ when no confusion will arise.