I am reading the paper On the Classification of Non-Compact Surfaces by Ian Richards. I have some questions about the statement of Theorem 3 on page 268.
$\textbf{Theorem 3.}$ Every surface is homeomorphic to a surface formed from a sphere $\Sigma$ by first removing a closed totally disconnected set $X$ from $\Sigma$, then removing the interiors of a finite or infinite sequence $D_1, D_2,...$ of nonoverlapping closed discs in $\Sigma\backslash X$, and finally suitably identifying the boundaries of these discs in pairs, (If may be necessary to identify the boundary of one disc with itself to produce an odd "cross cap.'') The sequence $D_1, D_2,...$ "approaches $X$" in the sense that, for any open set $U$ in $\Sigma$ containing $X$, all but a finite number of the $D_i$ are contained in $U$.
The questions are given below; I don't know why these are not mentioned in the paper, maybe obvious. But these appear while reading this paper first time, so it will be helpful if someone comments on my interpretation of this theorem.
Question 1: If I want only an orientable surface, then no need to identify the boundary of one disc with itself. Also, If I want a surface of the finite genus, say of genus $g$, then I need only $2g$ many discs $D_1,...., D_{2g}$ for pairwise identification. Is this right?
According to Theorem 2 on page 266, the space $X$ is homeomorphic to the space of ends. And some points of $X$ may give a planar end. Also, the space of planar ends corresponds to an open subset of $X$.
Question 2: If I want a surface with at least one non-planar end and at least one planar end, then the above bold line should be of the form: There is a closed subset $X_\text{np}$ of $X$ such that any open neighborhood (in $\Sigma$) of a point of $X_\text{np}$ contains infinitely many $D_i$, and each point of $\Sigma\backslash X_\text{np}$ has an open neighborhood (in $\Sigma$) which contains none of these $D_i$. Am I right?
Question: If my interpretation as in question 2 is right then why does the author consider arbtrary open set containing $X$ instead of considering arbitrary open neighborhood of each point of $X_\text{np}$?