A Taste of Topology

Abstract

It may seem paradoxical at first, but a specific math problem can be harder to solve than some abstract generalization of it. For instance, if you want to know how many roots the equation $t^5 - 4t^4 + t^3 - t + 1 = 0$ can have, then you could use calculus and figure it out. It would take a while. But thinking more abstractly, and with less work, you could show that any n ^th degree polynomial has at most n roots. In the same way many general results about functions of a real variable are more easily grasped at an abstract level — the level of metric spaces.