Topological Motivation

The idea of a metric allows us to apply analytical ideas to a variety of spaces. However, as Example 2 on Exercises 2 shows, different metrics sometimes lead to "the same analysis" (that is, to the same convergent sequences and consequently the same continuous functions). So, in a way, we demand too much information when we insist that we have a metric on a space.

It often happens that when we construct new spaces by "gluing things together" (like Möbius bands, real projective planes, complexes of various sorts) having to provide a metric on them is often difficult, sometimes impossible and always irrelevant.

Also, for some purposes, metric spaces do not let us do everything we want. For example, the idea of pointwise convergence of a sequence of functions in (say) C[0, 1] does not corespaond to convergence in any metric. To handle a situation like this needs a more general structure.