Is this meaning of "smallest" and "largest'" topologies wrong?

Question

One of Munkres' exercises has me prove "(...) there is a unique smallest topology, which contains all (...)", and "(...) a unique largest topology, contained in all (...)". The problem I'm having with the exercise is not proving that, I'm asking why the author uses "smallest" for "contains all topologies" and "largest" for "contained in all topologies". Should it not be the exact opposite? This definition would lead me to believe the indiscrete topology is "bigger" than the discrete topology, which is obviously false. What's going on here?

Answer 1

I believe you're misparsing the phrase. E.g.

"There is a unique smallest topology, which contains all ..."

does not mean "The unique smallest topology contains all ..." but rather "There is a unique (smallest topology which contains all ...)."

It's like saying "There is a unique smallest integer, which is bigger than $7$" - namely, $8$.

Answer 2

You are mis-parsing these phrases. "Smallest topology containing all the $\mathcal{T}_\alpha$" should be read as "smallest (topology which contains all the $\mathcal{T}_\alpha$)". That is, it is a topology $T$ which contains each $\mathcal{T}_\alpha$, such that if $T'$ is any other topology that contains each $\mathcal{T}_\alpha$, then $T\subseteq T'$.