Topology

Topology is the branch of mathematics that explores how shapes and spaces behave when they are stretched, bent, or deformed - as long as they’re not torn or glued. It focuses on features that stay the same under these transformations, such as continuity, connectedness, and the number of holes in an object.

What Makes Topology Unique

Unlike geometry, topology doesn’t care about exact distances or angles. Instead, it studies how things are connected and how one shape can be transformed into another without losing its fundamental structure. The key ideas here are the topological space - the framework where these relationships are defined - and the homeomorphism, the type of transformation that preserves them.

Sometimes called the “rubber-sheet geometry”, topology imagines objects as flexible surfaces that can stretch or twist freely, revealing the deeper connections between shapes.

How It All Began

The story of topology starts in the 18th century with Leonhard Euler and his famous Seven Bridges of Königsberg problem. Euler realised that the challenge of crossing each bridge once wasn’t about distances - it was about how the bridges connected parts of the city. This insight laid the foundation for the modern study of networks and connectivity.

In the 19th century, mathematicians like Gauss, Möbius, Riemann, and Klein deepened the field, while Johann Listing introduced the word “topology” itself. Later, Henri Poincaré brought a new level of abstraction, connecting topology with algebra and shaping the field as we know it today.

By the 20th century, topology had branched into several specialisations - including algebraic topology, differential topology, and general topology - influencing mathematics, physics, and even computer science.

Core Concepts Made Simple

A topological space is a way of describing how points relate to each other without using measurements. For example, the real number line \( \mathbb{R} \) is a simple topological space: it helps us define ideas like continuity and limits in a purely structural way.

A homeomorphism is a kind of “perfect reshaping” - a continuous transformation that can be reversed without breaking the object. If two shapes can be turned into one another this way, they’re topologically the same. That’s why a coffee cup and a doughnut are identical in topology: each has one hole and can morph into the other through smooth deformation.