5.2.1 Proof

We check that T_f satisfies conditions (T1)–(T3) for a topology.

Since (T1)–(T3) are satisfied, T_f is a topology on I(X).

Thus (I(X),T_f) is a topological space. We give the topology T_f a special name.

We now use F to create a topology on Y = f(X). We define a subset U of Y to be open if F⁻¹(U) is an open subset of X.

Recall that F([x]) = f(x).

So we have a bijection F:I(X) → Y between the topological spaces (I(X),T_f) and (Y,T_Y). Furthermore, from the definitions of open sets in T_f and T_Y, we can deduce immediately that for each U T_Y, F⁻¹(U) T_f and for each V T_f, F(V) T_Y. Hence F is a homeomorphism between (I(X),T_f) and (Y,T_Y).

We have thus shown that the object created by identifying some or all of the edges of a polygon is a topological space homeomorphic to the identification space of the polygon. We now need to show that this topological space is Hausdorff and that every point has a disc-like or half-disc-like neighbourhood, to demonstrate that the space is indeed a surface.