Triangulations

Abstract

The collection of all topological spaces is much too vast for us to work with. We have seen in previous chapters how to develop an abstract theory of topological spaces and continuous functions and to prove many important results. However, working in such a general setting we quickly run into two kinds of difficulty. On the one hand, in trying to prove a concrete geometrical result such as the classification theorem for surfaces, the purely topological structure of the surface (that it be locally euclidean) does not give us much leverage from which to start. On the other hand, although we can define algebraic invariants, such as the fundamental group, for topological spaces in general, they are not a great deal of use to us unless we can calculate them for a reasonably large collection of spaces. Both of these problems may be dealt with effectively by working with spaces that can be broken up into pieces which we can recognize, and which fit together nicely, the so called triangulable spaces.