The Bolzano-Weierstrass Theorem is an important result in point-set topology, the branch of mathematics that concerns the properties of sets and the points that comprise them. As an example, the real number line contains subsets of intervals, and the intervals contain real numbers, which are the points of this system. Thus we could consider the topology of the real number line to be the study of the properties of these intervals and the real numbers contained in them. We can also consider the topology of the Cartesian plane, of three-dimensional space, or of any higher dimensional spaces.

In order to understand the Bolzano-Weierstrass Theorem, it is necessary to understand four fundamental concepts: infinite sets, bounded sets, neighborhoods, and accumulation points. An infinite set is simply a set with an infinite number of points. The interval [0,1] is an infinite set in one dimension, as is a circle...