An Introductory Algebraic Topology

An important result in algebraic topology can very much be given by a powerful tool with which to attack certain types of problems. This is the equivalence of both the singular and the simplicial homology groups. The simplicial and the singular homology express the same ideas. The two constructions have different uses, as perhaps demonstrated by some of the examples and the theorems. In practice, it is much simpler to calculate homology groups by using the ideas of simplicial homology, but then, in many cases, it's easier and more straight forward in proving the theorems by using singular homology, as the notion of continuous maps is much more compatible with the second theory.