topological spaces 4

A closed set is a complementary set of the open set.
Therefore, we are able to define the topology by closed sets.

As with open sets, a collection of closed sets $\mathcal{F}_c\in\Omega$ 
has following properties.

(1) If $C_1,C_2\in\mathcal{F}_c$ , then $(C_1\cup C_2)\in \mathcal{F}_c$ .

(2) If $\left\{C_i(i=1,2,\cdots)\right\}$  is a collection of the elements of $\mathcal{F}_c$ ,
then $\cap_{i=1}^{\infty}C_i\in\mathcal{F}_c$ .

These are gotten by properties of open sets. The topology by closed sets
will be defined by adding next requirements.

(3) $\Omega,\phi\in \mathcal{F}_c$ .

The proof of (2) is based on the famous De Morgan's laws.

$(\cap C_i)^c=\cup C_i^c$

Please try the challenge.