Hello, I've two questions:

1) Given X a set then \mathcal{P}(X) is a topology on X off ( \mathcal{P}(X) being the power set of X).
Now if I take c and c \notin X is ( \{c \} \bigcup X ) \bigcup \mathcal{P}(X) still a topology on X? ( I'd say yes because the 3 axioms work... I think, I might be wrong...).

2) Let  \{ \mathcal{T}_{\alpha}\} be a family of topologies on X. Show that there is a unique smallest topology on X containing all the collections \mathcal{T}_{\alpha}.

for 2) I thought I could take the subbase  \bigcup_{ \alpha \in A} \mathcal{T}_{\alpha}. (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 \mathcal{T}_{\alpha} 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!!