Topology

Some shapes can change.

You can stretch or bend a shape. You can even twist it. But you must not tear it. You cannot make new holes. You also cannot glue parts together.

In this math, a donut and a cup are the same.

They both have one hole. A circle and a square are also the same. This is because they both have no holes.

One shape is a strip with only one side.

This is a special kind of shape. Math helps us see how things fit together. It is a very cool way to look at the world.

Imagine you have a piece of soft clay. You can stretch it or bend it. You can even twist it into a new shape. But you must not tear it. You cannot glue parts together or make new holes.

This way of looking at shapes is called topology. In topology, some shapes are considered the same. A circle and a square are the same. They both have no holes. A coffee mug and a doughnut are also the same. They both have exactly one hole. This special link is called a homeomorphism.

Topology helps us see how things are put together. Long ago, Leonhard Euler studied a town called Königsberg. The town had seven bridges. He wanted to know if a person could cross every bridge just once.

He found it was impossible. His work did not care about the length of the bridges. It only cared about how they connected. This led to new math called graph theory. Another shape is the Möbius strip. It is a special strip with only one side.

Imagine you have a piece of soft, stretchy clay. You can pull it, twist it, or bend it into many new forms. However, you must follow two strict rules. You cannot tear the clay to make a new hole. You also cannot glue parts together to close an existing hole.

This way of studying shapes is called topology. In topology, we do not care about exact measurements like length or width. Instead, we look at how an object is put together. A circle and a square are seen as the same. This is because they both have no holes.

When two shapes can be changed into each other without tearing or gluing, they are called homeomorphic. This means they share the same topological properties. A famous example is the "Topologist's Breakfast." To a topologist, a coffee mug and a doughnut are the same shape. You could slowly push a dimple into a doughnut to make a mug handle. The central hole of the doughnut simply becomes the hole in the handle.

Other properties include dimension and connectedness. Connectedness helps us tell the difference between one circle and two separate circles. Compactness helps us distinguish a simple line from a circle.

History shows that these ideas grew slowly over many centuries. In the 1700s, Leonhard Euler studied a famous puzzle. It was called the Seven Bridges of Königsberg.

He wanted to know if someone could walk across every bridge exactly once. Euler found this was impossible. He did not need to know the length of the bridges to solve it. He only needed to see how they connected the land. This work helped start a new field called graph theory.

Many thinkers helped build this math field. Johann Benedict Listing used the word "topology" in the 1800s. Later, Henri Poincaré made huge advances in the 1890s. He introduced ideas like homotopy, which involves squishing objects. In 1906, Maurice Fréchet introduced metric spaces. In 1914, Felix Hausdorff defined what we now call a topological space.

These mathematicians turned simple observations into a deep science. They moved from looking at single shapes to studying entire spaces.

Topology connects to many things we see every day. You can see it in the way a Möbius strip works. A Möbius strip is a special shape with only one side and one edge.

You can also see topological ideas in the "hairy ball theorem." This theorem says you cannot comb the hair on a ball flat without making a cowlick. This rule applies to any smooth blob that has no holes. Whether it is a sphere or a lumpy shape, the math stays the same. Topology shows us the hidden rules that shapes must follow.

Topology is a specialized branch of mathematics focused on the properties of geometric objects. These properties remain unchanged during continuous deformations. Such deformations include stretching, twisting, bending, or crumpling an object. However, these transformations must follow strict rules. One cannot tear the object to create new holes. One also cannot glue parts together to close existing holes.

In this field, the exact size or shape of an object matters less than how it is connected. A circle and a square are considered the same because they are both one-dimensional objects that separate a plane into an inside and an outside.

To describe these transformations, mathematicians use specific terms. A homeomorphism is the most basic form of topological equivalence. Two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A famous illustration of this is the "Topologist's Breakfast." In this example, a coffee mug and a doughnut are considered identical. A pliable torus, which is a doughnut shape, can be reshaped into a mug. One simply creates a dimple that grows into the mug's body while the central hole becomes the handle.

Another concept is homotopy equivalence. This occurs when two objects result from "squishing" a larger object.

Topologists look for properties that are invariant under these deformations. These are called topological properties. One such property is dimension, which distinguishes a line from a surface. Another is compactness, which helps distinguish a line from a circle. Connectedness is also vital, as it allows us to tell a single circle apart from two non-intersecting circles.

Other advanced concepts include orientability and Betti numbers. These help define the complex characteristics of manifolds, which are spaces that look like Euclidean space near every point.

The history of topology involves several centuries of discovery. In the 18th century, Leonhard Euler provided one of the field's first practical applications. He studied the Seven Bridges of Königsberg problem in 1736.

He proved it was impossible to cross all seven bridges exactly once. This result did not depend on bridge length or distance. It only depended on connectivity properties. This discovery helped lead to the development of graph theory. Euler also developed a polyhedron formula involving vertices, edges, and faces.

As the field grew, the formal terminology emerged. Johann Benedict Listing introduced the term "topology" in the 19th century. In 1895, Henri Poincaré published his work on Analysis Situs. This paper introduced the concepts of homotopy and homology, which are parts of algebraic topology. In the early 20th century, the idea of a topological space was fully developed. Maurice Fréchet introduced the metric space in 1906. Later, Felix Hausdorff coined the term "topological space" in 1914.

Kazimierz Kuratowski further generalized these ideas in 1922.

Topology reveals surprising truths about the world. One example is the hairy ball theorem from algebraic topology. This theorem states that one cannot comb the hair flat on a hairy ball without creating a cowlick. Formally, this means there is no nonvanishing continuous tangent vector field on a sphere. This rule is not about the sphere's exact shape. It applies to any smooth blob that lacks holes. Similarly, the Möbius strip is a unique topological object. It is a surface that has only one side and one edge.

Modern topology is a vast discipline with many subfields. General topology, or point-set topology, deals with basic set-theoretic definitions. It uses concepts like open sets, closed sets, and neighborhoods. This serves as the foundation for differential and geometric topology. The field is deeply connected to set theory, which was developed by Georg Cantor. Today, topology remains a vital area of study. Its importance was recognized when Dennis Sullivan won the 2022 Abel Prize for his contributions to its many aspects.