Let $X$ be a set. Suppose $\beta$ is a basis for the topology $\tau_\beta$ of $X$. Since each base element is open (with respect to $\tau_\beta$) we have that $B\in \beta\Rightarrow B\in \tau_\beta.$ Thus, $\beta\subset \tau_\beta$.
However, since $\beta$ is a union of base elements (I assume a set can always be written as a union of its elements) and topologies are closed under arbitrary unions, we have $\beta \in \tau_\beta.$ Is it possible for a set to be a subset an an element of another set?