Does the end of the 2-sliced Loch Ness Monster surface have a fundamental system of neighbourhoods $(V_i)$ such that the frontier of $V_i$ is an arc?

Question

Let $S$ be the $2-$sliced Loch Ness Monster surface. Does it end have a fundamental system of neighbourhoods (local base) $(V_i)$ such that the frontier of each $V_i$ is an arc? I was reading a paper by John Mather, "Invariant subsets for area preserving homeomorphisms of surfaces", where he says that every end of a surface with boundary has a fundamental system of neighbourhoods $(V_i)$ such that the frontier of each $V_i$ is a circle or arc. Since $\partial S$ has two connected components, I can't see how this is possible in the case of $S$.

Answer 1

Here's how to do it with a closed arc; sort of a weird neighborhood base in that it's gonna have a slice in it, but it'll do the job. If you meant open arc, then I guess you could just add a little bit on at the ends. To do it with the circle is similar.

Just to be clear, the neighborhood is the unbounded component of the complement of the red curve. You can do it at each handle.