topological spaces 3

If $A_1$  and $A_2$  are open sets in $\Omega$, then $(A_1\cap A_2)$  is a open set. We have already proved it in "open sets 2(the intersection is open)" generally.

If $(A_1\cap A_2)$  has no elements, or $A_1$  is equal to $A_2$ ,
then the proof would even be clear.

In the case which $(A_1\cup A_2\cup\cdots )$  is open, the proof would not also be required.

Therefore, the definition of a topological space essentially gives the collection of open sets in $\Omega$ .

Precisely the collection of open sets is called the topology,
and $\Omega$  is called the support or the set theoretic support.

For example, as $(\Omega, \phi)$  is the collection of open sets,formally $(\Omega, \phi)$  gives a topology.
Let $\left\{ (-\infty,0)\cup (0,\infty)\right\}$  be $\Omega$ . $(-\infty,0)$  and $(0,\infty)$  give a topology, too.

That is to say, on a $\Omega$ , various kinds of a toplogical space can be defined.