# Powerful Topology Theorems You Should Be Aware Of

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## Introduction to Topology

Did you know that there is always at least one spot on Earth where the wind is completely still at any given moment? Or that there are two points on the planet's surface that share the same temperature and barometric pressure? These intriguing facts can be demonstrated using mathematics, specifically through the principles of algebraic topology.

Algebraic topology provides us with broad insights, revealing that certain properties must be met without detailing how to locate them. Among the most significant results is the Brouwer Fixed Point Theorem.

### Formal Presentation of the Brouwer Fixed Point Theorem

The Brouwer Fixed Point Theorem, a pivotal result in algebraic topology established in the early 20th century, underpins numerous vital theories across various disciplines, including economics, game theory, chemistry, and mathematics itself.

Here’s a formal definition:

Brouwer Fixed Point Theorem: For any continuous function f mapping a nonempty compact convex set to itself, there exists a point x such that f(x) = x.

This point is termed a fixed point. It’s essential to understand that this version of the theorem does not specify the elements of the set. Typically, it refers to a subset of Euclidean space in any dimension. A compact space means it is closed and bounded, encompassing its boundary (limit points), and has a finite area. Convexity implies that any line segment joining two points within the space lies entirely inside it, ensuring connectedness.

A classic example of such a space is a disk or an n-dimensional ball. Interestingly, the theorem also holds true for any set homeomorphic to a closed ball, which must also be closed, bounded, and connected, though not necessarily convex.

The significance of this theorem lies in its ability to inspire an entirely new field of mathematics and various fixed-point theorems, with the Brouwer theorem often regarded as the foundational one. For instance, John Nash utilized it to establish the existence of his renowned Nash Equilibrium, which earned him the Nobel Prize.

To illustrate the theorem's practical application: consider two sheets of paper with identical images, one placed directly atop the other. If you crumple the top paper and lay it over the other, the Brouwer Fixed Point Theorem guarantees that at least one point on the crumpled paper aligns directly above the corresponding point on the bottom sheet, regardless of how the top paper is deformed—provided it remains intact.

It's an intriguing exercise to attempt to prove this in specific cases or even the complete scenario if you're up for the challenge. In one dimension, for instance, it simplifies to the intermediate value theorem.

### Illustration of the Brouwer Fixed Point Theorem

## The Hairy Ball Theorem

Mathematicians have expressed the Hairy Ball Theorem with the phrase: "You can’t comb a hairy ball." This essentially means you cannot comb it flat. The theorem, first rigorously proven by the eminent mathematician Henri Poincaré in 1885, has since been generalized.

Why discuss hairy balls, you might wonder? While they may seem trivial, proven mathematical results often yield valuable applications. Before delving into its significance, let’s state the theorem formally:

Hairy Ball Theorem: There is no non-vanishing continuous tangent vector field on any even-dimensional sphere.

Here, n is a natural number, making 2n any positive even number. This theorem indicates that if a continuous tangent vector field exists on such a sphere, at least one vector must equal zero.

To clarify, a continuous tangent vector field assigns a vector to each point in an n-dimensional Euclidean space. These tangent vectors can be thought of as vectors that point "along" a surface, and continuity implies that small distances between points correspond to small distances between the associated vectors.

If we envision a flow on a two-dimensional sphere, the Hairy Ball Theorem asserts there will always be at least one point where the flow is zero, or in other words, where speed is null.

This concept also applies to air as a fluid, suggesting that the theorem holds for various dimensions and even in theoretical contexts. The true strength of mathematics lies in its ability to generalize and abstract, enabling us to comprehend even the most complex concepts.

### Applications of the Hairy Ball Theorem

## The Borsuk-Ulam Theorem

The Borsuk-Ulam Theorem is among the most fascinating and unexpected theorems in mathematics. It asserts that there are always two antipodal points on the Earth's equator that have identical temperatures, or two points on the surface that share both temperature and pressure. This is a universal truth, applicable at any moment.

The full theorem states:

Borsuk-Ulam Theorem: Every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

It’s intriguing to note that we can establish this without necessarily identifying those two equivalent points under the function, although modern techniques can facilitate finding such points. This means that irrespective of how we manipulate values on the sphere, as long as continuity is maintained, two such points with identical values will always exist at any given moment.

These three theorems—Brouwer Fixed Point, Hairy Ball, and Borsuk-Ulam—are foundational to algebraic topology. Although there are many more intriguing theorems, this selection highlights their significance and potential applications.

You now possess knowledge about phenomena related to Earth's weather that even meteorologists may overlook. Thank you for engaging with this exploration of topology!

### The Ham Sandwich Theorem and Continuum

This video explores the Ham Sandwich Theorem and its implications in algebraic topology, shedding light on its mathematical significance.

### Introduction to the Fundamental Group

In this video, you will learn about the fundamental group in algebraic topology, providing a solid foundation for understanding the subject further.