Let $X$ be an infinite set and let $C(X)$ denote the collection of connected Hausdorff topologies on $X$. Suppose $N\subseteq C(X)$ has the property that whenever $\tau\neq\sigma \in N$ then $(X,\tau)$ and $(X,\sigma)$ are not homeomorphic. In terms of $|X|$, how large can $|N|$ be at most?



Cardinality of connected Hausdorff topologies
Let $\kappa:=|X|$. Then $2^{2^\kappa}$ is an obvious upper bound for the number of topologies on $X$. 


Every ultrafilter on $\kappa$ will give you a Hausdorff topological space on $\kappa+1$; these are $2^{2^\kappa}$ many spaces. Some of them might be homeomorphic, but there are only $2^\kappa$ many bijections, so you get $2^{2^\kappa}$ many non-homeomorphic spaces. 


Assuming that $\kappa$ is at least continuum, you can add one more point, and copies of the unit interval to each point of your original space; this will now give you a connected (even path-connected) space. 


If $\kappa$ is smaller than continuum, then replace the unit interval by some countable connected Hausdorff space. 


