Chapter 2.17, Problem 1E

Let $\mathcal{C}$ be a collection of subsets of the set $X$. Suppose that $\varphi$ and $X$ are in $\mathcal{C}$, and that finite unions and arbitrary intersections of elements of $\mathcal{C}$ are in $\mathcal{C}$. Show that the collection

$\mathcal{T}=\left\{X-C|C\in \mathcal{C}\right\}$

is a topology on $X$.