1 (a)

Let $\displaystyle \mathcal{C}$ be a collection of subset of the set X, satisfying:

i. $\displaystyle \phi$ and X contained in $\displaystyle \mathcal{C}$,

ii. finite unions of elements in $\displaystyle \mathcal{C}$ are in $\displaystyle \mathcal{C}$, and

iii. arbitrary unions of elements in $\displaystyle \mathcal{C}$ are in [$\displaystyle \mathcal{C}$

Show that the collection

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

is a topology on X.

(b)

Give an example of such a topology when X = $\displaystyle \Re$. What are the open sets in this topology?What are the closed sets? What is the basis for the topology?