A topological space is a pair $(X, \tau)$ where $X$ is a set and $\tau \subset \mathcal P(X)$ is a collection of subsets of $X$ such that:
\begin{align}
\varnothing &\in \tau, \\
X &\in \tau,
\end{align}
for every family $(U_i)_{i \in I}$ with $U_i \in \tau$ for all $i \in I$,
\begin{align}
\bigcup_{i \in I} U_i &\in \tau,
\end{align}
and for every finite family $U_1,\ldots,U_n \in \tau$,
\begin{align}
\bigcap_{i=1}^n U_i &\in \tau.
\end{align}
The elements of $\tau$ are called open sets.