topological spaces

We shall give the standard definition of a topological space.

For a set $\Omega$ , given $\mathcal{F}$ which is the family of subsets of $\Omega$ 
(it is a element of the power set of $\Omega$) .
$\Omega$  is a topological space, if following conditions are satisfied;

(1)$\Omega\in \mathcal{F}$ , and $\phi\in \mathcal{F}$ .
(2)If $A_1\in \mathcal{F}$ and $A_2\in \mathcal{F}$, then $A_1\cap A_2\in \mathcal{F}$ .
(3)If $\left\{A_i(i=1,2,\cdots )\right\}$  is a family of the elements of $\mathcal{F}$ 
(namely, all $A_i$ is the element of $\mathcal{F}$), then $\cup_{i}A_i\in \mathcal{F}$ .

We also say $A_i$  a open set of $\Omega$ . 

It is equivalent to the preceding definition(coffee break 8-2).
However, the definition of a open set is given additionally.