Hello, I've two questions:

1) Given X a set then is a topology on X off ( being the power set of X).

Now if I take c and is still a topology on X? ( I'd say yes because the 3 axioms work... I think, I might be wrong...).

2) Let be a family of topologies on X. Show that there is a unique smallest topology on X containing all the collections .

for 2) I thought I could take the subbase . (Where A is a set), but if I'm right in 1) then I think there is a problem bc nothing tells me that none of the is not of the form of what I wrote in 1) so then it wouldn't be a subbase anymore.

So can someone give me a hit for 2) (or write the whole thing if you have time).

thanks in advance!!