The standard topology on $\mathbb{R}$ (real number system) is the topology $T$ whose open sets consist of unions of open intervals in $\mathbb{R}$. That is, $U \subseteq \mathbb{R}$ is open if, for each $x \in U$, there exists an open interval $(a,b)$ such that $x\in (a,b)$. Note that $\emptyset \in T$ trivially.
Could you explain what standard topology is. I can understand the last sentences but I have a problem with the first sentence. What does "open sets consist of unions of open intervals in $\mathbb{R}$"? Could you write what the topology is equal to? Thank you.
The open sets consists of unions of SETS OF open intervals. That is, $S$ is open iff there exists a set $C$ of open intervals such that $S=\cup C=\{r: \exists c\in C\;(x\in C)\}. $
A 'union of open intervals" is a commonly-used abbreviation for this.