Here, I’ll discuss a famous fixed-point theorem from topology.

Definition and Statement of the Theorem

A fixed point is literally a point that moves to itself when moved by a continuous mapping. Formally, let \( X \) be a topological space, and \( f:X→X \) a continuous map. A point \( x \) that satisfies \( f(x)=x \) is called a fixed point of \( f \).

Brouwer’s Fixed-Point Theorem states that

let \( D^n \) is an \( n \)-dimensional disk and \( f:D^n \to D^n \) is a continuous map, then \( f \) must have at least one fixed point.

Intuitive Examples for Lower Dimensions

Let’s try to intuitively understand this for lower dimensions.

• When \( n=1 \): \( D^1=[−1,1] \). If we think of a curve \( f \) connecting opposite sides of a square, it must intersect the diagonal \( x=f(x) \). This case can also be proved using the intermediate value theorem.
• When \( n=2 \): \( D^2 \) is a regular disk. Suppose you have two pieces of paper with coordinates drawn on them. Leave one piece of paper as is, crumple up the other piece of paper, and stack the two pieces of paper on top of each other (You can also shrink the other piece of paper and stack it on top of the other piece of paper).
• When \( n=3 \): \(D^3 \) is an ordinary sphere. If the liquid in the container is stirred, there will be at least one point after the stirring that does not change from the original position. Of course, we assume the map from the original point to the point after stirring is continuous.

Applications and Generalizations

There are many fixed-point theorems in mathematics. For example, Kakutani’s fixed-point theorem, which is a generalization of Brouwer’s fixed-point theorem, appears in economics. The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics. This work later earned Nash a Nobel Prize in Economics.