$\mathcal T$ is a collection of arbitrary union of elements of the base

Question

Suppose a group $X$ with a base $\mathscr B$ for topology $\mathcal T$. Show that $\mathcal T$ is a collection of arbitrary union of elements of $\mathscr B$.

I know the definition of topology and the definition of base but how can I show that? any hints?

Answer 1

Let $U\in\mathcal T$.

Then for every $x\in U$ some $B_x\in\mathscr B$ exists with $x\in B_x\subseteq U$.

Consequently: $$U=\bigcup_{x\in U}B_x$$