I'm doing this exercise:
Let $\{T_\alpha\}$ be a family of topologies on $X$. Show that there
is a unique smallest topology on $X$ containing all the collections
$T_\alpha$, and a unique largest topology contained in all $T_\alpha$.
I have proved everything except the unique part.. I just can't get my head around what is meant with unique here. Which may sounds silly.
I have proved that the intersection is a topology. And if you are a topology that is also contained in every $T_\alpha$, than you surely are contained in the intersection, so you are not larger.
But I don't see from what it follows that this intersection is the unique largest topology contained in all $T_\alpha$. One part of my head say it is trivial, the other part gets confused. Like it is redundant to talk about unique in this context.
The same for proving the uniqueness of the smallest topology.
Edit Should I read topology $A$ larger than topology $B$ as, $A$ has more elements than $B$ ? I thought that, because the author uses the word finer for $A \supset B$.
Best Answer
If a topology is largest, it's unique automatically. Suppose $\tau_0$ and $\tau_1$ were both largest topologies. Then $\tau_0\supseteq\tau_1$ and $\tau_1\supseteq\tau_0$, hence $\tau_0 = \tau_1$.
Maximal, on the contrary would not have to be unique.