5.3.3 Torus with 1 hole
In our last example, we consider a pentagon with two pairs of edges identified. As we saw in Section 2.3, identification of the edges produces a torus with a hole. In this case there are five vertexneighbourhoods to fit together, as shown in Figure 109. Identification of edges a and b shows that neighbourhoods 2, 3 and 4 fit together as shown. Identification of the edges b shows that neighbourhood 5 fits alongside neighbourhood 2, and identification of edges a shows that neighbourhood 1 fits alongside neighbourhood 4. However, neighbourhoods 1 and 5 do not fit together, and so we do not obtain a disclike neighbourhood – rather we obtain a halfdisclike neighbourhood and the resulting vertex lies on the boundary of the hole.
Finally consider a point x that lies on the edge c of the pentagon (see Figure 110). This has a halfdisclike neighbourhood. After identification of edges, this point lies on the boundary of the hole, and maintains its halfdisclike neighbourhood.
From these examples we can deduce that:

if a point lies on an edge that is identified with another edge, but not at a vertex of that edge, then after identification that point has a disclike neighbourhood;

if a point lies on a part of the boundary that is not identified with another part, then after identification that point has a halfdisclike neighbourhood.
However, if a point is vertex of an edge that is identified with another edge, we have seen that after identification that point can have a disclike or halfdisclike neighbourhood. We need to be sure that these are the only possibilities.
Consider the vertices x_{1}, x_{2}, …, x_{k} of a polygon that are identified to the same point x by the edge identifications. In the polygon, each vertex x_{i} has a halfdisclike neighbourhood that can be represented as in Figure 111. We shall call such a piece a wedge.
Some of the edges that come into each vertex x_{i} are to be identified. Let us see what can happen. First, notice that the edge identifications that involve the vertices x_{i} cannot have wedges that fit together to form two or more separate pieces: this is because, in such a case, the edge identifications would not identify all the vertices x_{i} to a single point x. Also, because whole edges and not just vertices are identified, we cannot produce a situation such as that illustrated in Figure 112. Thus, the edge identifications that involve the vertices x_{i} are such that all the wedges fit together to form a single neighbourhood. Either the point x is in the interior of the object so formed, as for the torus and Klein bottle, giving rise to a disclike neighbourhood formed from the wedges, or the point x lies on the boundary, as for the torus with 1 hole. Thus:

if a point lies at a vertex of an edge that is identified with another edge, then after identification that point has a disclike or halfdisclike neighbourhood.
We have therefore demonstrated that every point of polygon with edge identifications has, after identification, a disclike or halfdisclike neighbourhood. Furthermore, given any two points of the object obtained after identification, by choosing disclike or halfdisclike neighbourhoods of sufficiently small radius, we can find neighbourhoods that do not intersect – so the object is Hausdorff. Combining these results with those of Section 5.2 tells us that an object formed by identifying edges of a polygon is a surface, as defined in Section 2.5. Furthermore, we know that such a surface is homeomorphic to the identification space of the polygon with edge identifications.