After thinking about this [question](https://mathoverflow.net/questions/24932/counting-connected-manifolds/24938#24938) and reading this [one](https://mathoverflow.net/questions/4155/classification-problem-for-non-compact-manifolds/11259#11259) I am led to ask for an uncountable collection of homeomorphism types of boundaryless connected path-connected submanifolds of the plane. 


My guess is that is suffices to consider complements of Cantor sets. However, I do not know how to distinguish ends (up to homeomorphism) sufficiently well to ensure that this works. Are there other, easier, invariants?



Counting submanifolds of the plane
See a [theorem of Richards](http://www.ams.org/mathscinet-getitem?mr=143186), which implies that homeomorphism types of planar surfaces are in 1-1 correspondence with homeo. types of compact subsets of the Cantor set. I think there should be uncountably many homeo. types of totally disconnected compactums, but I don't know a reference or an argument off the top of my head. I think this should be related to the ordinalities of the accumulation points, but I'm not sure which ordinals can occur. 


Addendum: Googling, I found references to a result of Markiewicz-Sierpinski classifying
countable compact metric spaces up to homeomorphism by their [Cantor-Bendixson rank](http://en.wikipedia.org/wiki/Derived_set_%28mathematics%29#Cantor.E2.80.93Bendixson_rank) (see [section 3 of this paper](http://arxiv.org/abs/0710.1495) for a statement). 
The CB-rank must be a countable [ordinal](http://en.wikipedia.org/wiki/Ordinal_number) $\zeta$, and the space is homeomorphic to the
ordinal $\omega^\zeta\cdot n+1$ with the order topology for some $n\in \mathbb{N}$. These may all be realized as
compact subsets of the line. This gives uncountably many non-homeomorphic compacta, 
which by Richards' theorem implies that there are uncountably many planar surfaces. 


