I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post.


I am reading this [thesis](https://trace.tennessee.edu/cgi/viewcontent.cgi?article=8804&context=utk_graddiss).


Corollary 4.1.15. on page 63 says that if the number of ends (see Definition 4.1.6. of the above thesis) of Freudenthal space (i.e., a $\sigma$-compact, locally compact, connected, and locally connected space) is infinite, then its space of ends is homeomorphic to the Cantor set.


As far as know, the space of ends of a non-compact surface can be any closed subset of the Cantor set; see Theorem 2 of [this](https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143186-0/S0002-9947-1963-0143186-0.pdf). For example, the space of ends of $\Bbb R^2\setminus \Bbb N$ is homeomorphic to the space $\{1/n\}\cup\{0\}$, which is an infinite set but certainly not homeomorphic to the Cantor set.


So, my question is the following:



> 
> Are these two notions of the space of ends (one coming from
> Freudenthal compactification and another, as mentioned in Ian Richard's
> paper) different?
> 
> 
> 



If the number of ends of Freudenthal space is infinite, then its space of ends is homeomorphic to the Cantor set?
Of course, yourself and YCor already answered this in the comments, but since I see you tagged geometric group theory, maybe you will be interested in this explicit answer about bi-Holder homeomorphisms : The end boundary of an accessible infinitely-ended group is bi-Hölder equivalent to the standard Cantor ternary set if and only if it is virtually free. This is Corollary 1.9 of this paper : <https://arxiv.org/abs/2010.07671>. The involved distance is the visual distance with respect to a base-point.


