Show the quotient space of a finite collection of disjoint 2 simplices obtained by identifying pairs of edges is always a surface, locally homeomorp

Question

Show the quotient space of a finite collection of disjoint 2-simplices obtained by identifying pairs of edges is always a surface, locally homeomorphic to $\mathbb{R}^2$.

I have thought about doing the following: I think we have to consider several cases

To prove that this space is a surface, we must take a point and prove that there is an open that contains it that is homeomorphic to the plane, if the point belongs to the interior of a 2-simplex that this space includes, we are ready the open is 2-simplex itself, the problem is if the point in question belongs to the intersection of two or more 2-simplices, how can I do in this case to be well defined? Thank you!

Edit: This question is part of the exercises in Hatcher's book, in particular, exercise $10.(a)$ (pag 131), the complete exercise is:

Note that: Each edge is identified with exactly one other edge.

Answer 1

See John Lee's Introduction to Topological Manifold (e2) proposition 6.4 (b) for local homeo, and 5.23 and 6.4 (a) for hausdorff and 2nd countable.

Answer 2

One must demonstrate an atlas of coordinate charts with continuous transitions. For any point on the interior of a polygon, this is trivial. For other points, the strategy is to draw circles around the point and traverse through identifications.

For a point on an edge of the polygon that is not a vertex, you can capture a neighborhood by drawing a semicircle that does not go around any vertices, then following the gluing and drawing another semicircle. The enclosed region is evidently homeomorphic to a neighborhood of $\mathbb{R}^2$.

Around verticies, you do essentially the same thing: you draw segments of circles, following identifications around until you're back to where you started. Note that this process is well-defined as every edge is glued to exactly one other edge, and must terminate in finitely many steps since each edge only gets crossed once. Mapping the enclosed neighborhood into a disk in the plane (reparametrizing angles so as to match up at the end) then gives a chart around the vertex. A bit of thought shows that the charts these describe can be made to have continuous transitions.