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**redsoxfan325** Let $\displaystyle X$ be an infinite set and let $\displaystyle T=\{\mathcal{T}~|~\mathcal{T}~\mbox{is~a~topology~ on~} X\}$

How big is $\displaystyle |T|$?

What if we restrict $\displaystyle T$ to be the set of all the topologies on $\displaystyle X$ up to homeomorphism?

It's pretty clear that an upper bound (in either case) is $\displaystyle |2^{2^X}|$. In the first case, $\displaystyle |X|$ is definitely a lower bound, because for each $\displaystyle x\in X$, we can define $\displaystyle \mathcal{T}_x=\{\emptyset,\{x\},X\}$. In the second case, it's got to be infinite, because you can take a countable subset $\displaystyle \{x_1,x_2,...\}$ and define the topologies $\displaystyle \mathcal{T}_1=\{\emptyset,\{x_1\},X\}$ and $\displaystyle \mathcal{T}_n=\mathcal{T}_{n-1}\cup\{U\cup\{x_n\}~|~U\in\mathcal{T}_{n-1}\}$.

Beyond this, I'm not too sure.