It is the power set of $\Bbb R$ a topology of $\Bbb R$? Maybe this is a very easy question of topology but I dont have all things clear. 
The question is: the whole power set of $\Bbb R$ is a topology of $\Bbb R$ or not?
The axioms to define a topology $\tau$ in a set X are:


*

*$\emptyset,\ X \in \tau$

*Every finite intersection of elements of $\tau$ belongs to $\tau$ too

*Every union of elements of $\tau$ belongs to $\tau$ too


If $\mathscr P(\Bbb R)=\tau \ $ I cant see a way where axioms dont hold but this means that all subsets of $\Bbb R$ are open and closed.
It is correct my reasoning or this is wrong? Thank you in advance.

It is the largest topology on $\mathbb R $: every set is open (and thus every set is closed). In this topology, any function $f:\mathbb R\to T $, for any topological space $T $, is continuous.
The other extreme is the topology $\{\mathbb R,\emptyset\} $, in which no nontrivial set is open, and so the only continuous functions are the constants.
