I am trying to show the co-countable topology $\tau$ is a topology on a set $X$.
The first point to show $\emptyset,X\in\tau$ is not difficult.
Now let $S_i\in\tau$. From the definition of co-countable, $S_i=\emptyset$ or $X\backslash S_i$ is countable.
I struggle to prove the second point $\cup S_i\in\tau$:
If $S_i=\emptyset$, then $\cup S_i=\cup\emptyset=\emptyset\in\tau$. The part I got stuck is what if $S_i\neq\emptyset$? So far I have not used $X\backslash S$?
The thing that troubles me a lot is that we are not told beforehand whether $X$ is countable or uncountable. Is it necesary to know?
How are we going to work on the condition being countable?
The third point $\cap S_i\in\tau$ should be similar but I am also not very sure about how can we make use of $X\backslash S_i$?
Thanks for any help on this matter.