Hello, I've two questions:

1) Given X a set then $\displaystyle \mathcal{P}(X)$ is a topology on X off ( $\displaystyle \mathcal{P}(X)$ being the power set of X).

Now if I take c and $\displaystyle c \notin X$ is $\displaystyle ( \{c \} \bigcup X ) \bigcup \mathcal{P}(X)$ still a topology on X? ( I'd say yes because the 3 axioms work... I think, I might be wrong...).

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

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