# Math Help - Empty Set as a union of Basis Elements

1. ## Empty Set as a union of Basis Elements

I just had a quick question.

If $\mathcal{B}$ is a basis for a topology on a set $X,$ defined in the way

1) $X = \displaystyle\bigcup_{B \in \mathcal{B}} B$
2) If $B_1, B_2 \in \mathcal{B},$ and $x \in B_1 \cap B_2$, then there exists $B_3 \in \mathcal{B}$ such that $x \in B_3 \subseteq B_1 \cap B_2$

Then it is a fact that this generates a topology on $X$, where the sets in the topology are unions of elements of $\mathcal{B}$. But we require $\emptyset$ to be an element in this topology - is it ok to take a union of none of the basis elements?

Thanks for any help.

2. ## Re: Empty Set as a union of Basis Elements

By definition $\big\cup\{B_i:i\in\emptyset\}=\emptyset$ .