Given space $X = \{a, b, c\}$, $\beta$ is a basis for a topology $\tau$ on X.
$\tau = \{ \varnothing, X, \{a\}, \{b\}, \{a,b\}\}$, $\beta = \{\{a\}, \{b\}, X\}$.
$\beta$ can't union its elements to get empty set $\varnothing$ contained in $\tau$ , but the definition of basis require that every open set can be expressed as a union of basis elements.
So why $\varnothing$ is not an element of $\beta$ ?
Best Answer
The definitions may vary a bit from place to place, but note that $\varnothing$ is the union of no element from the basis. That is to say, there is some $A\subseteq\beta$ such that $\bigcup A=\varnothing$.
If you require that "for all $U\in\tau$ there is $A\subseteq\beta$ such that $U=\bigcup A$", then this fine.
(Also note that some people tend to neglect the empty set, because we already know it has to be there, e.g. "cofinite topology" is all the cofinite sets and the empty set, which need not be cofinite. People often omit, or forget, the empty set. But it's fine because we know it has to be there anyway.)