Borsuk-Ulam theorem

The Borsuk-Ulam theorem is one of the most important and profound statements in topology: if there are n regions in n-dimensional space, then there is some hyperplane that cuts each region exactly in half, measured by volume. All kinds of interesting results follow from this. For example, at any given moment on the Earth's surface, there must exist two antipodal points – points on exactly opposite sides of the Earth – with the same temperature and barometric pressure! One way to see that this must be true is to think of two opposite points A and B on the equator. Suppose A starts out warmer than B. Now move A and B together around the equator until A moves into B's original position, and simultaneously B into A's original position. A is now cooler than B, so somewhere in between they must have been the same temperature. The Borsuk-Ulam theorem implies the Brouwer fixed-point theorem and also the ham sandwich theorem.