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.
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. 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?