We know that a polygon X is a surface and so each point x in X has a disc-like or half-disc-like neighbourhood. We shall show that a map f that identifies edges of a polygon to create an object Y automatically creates corresponding disc-like or half-disc-like neighbourhoods of each point y = f(x) of Y.
If x is in the interior of X, there is no difficulty: the point x has a disc-like neighbourhood U which is mapped by f to a disc-like neighbourhood f(U) of y = f(x) in Y.
If x lies on the boundary of X, there are three cases to consider:
x lies on an edge that is identified with another edge, but not at a vertex of that edge;
x lies at a vertex of an edge that is identified with another edge;
x lies on a part of the boundary that is not identified with another part.
In each case, x has a half-disc-like neighbourhood in X. Some examples will help us determine which happens to these half-disc-like neighbourhoods after identification takes place.