What Is Topology And Why You Should Love It
Examples of topological problems and real-life applications
I love topology because it always felt to me like “vibe math”. Coming to study advanced math without any previous background in that, the topological courses were the ones where I was able to get even the highest grade (as opposed to most other courses, haha), even without previous study. I could just vibe-udesrtand the concepts. It’s visual, a lot, it seems intuitive. If you have numbers and formulas but like to think and be abstract, it might be your favorite part of math, too.
This blog post is about what topology is and what applications it has. You will learn key topological concepts, and I will give you interesting resources to check.
So What Is Topology?
If you’ve ever heard someone try to explain topology, chances are they said something like:
“Topology studies properties that don’t change under continuous deformation.”
Or:
“Two objects are equivalent if you can morph one into the other without cutting or gluing.”
That’s, of course, technically correct, but if you’re like me, you want more than abstract definitions, more than just the infamous sketchy intuition about a donut turning into a mug in topology.
Most people are used to doing math on vector spaces or with numbers, where there’s already a lot defined - you can count, sum, measure, orient, and project. But not every part of math needs all of that.
I will always tell you that it’s nice to see math as a bunch of mathematical structures. There are many kinds of structures a set or space can have. Depending on which properties you define, you get a different world.
Some of the common mathematical structures:
Geometric structure - You care about angles, distances, and proportions.
Relational structure - You just define relationships between elements. For example, in graphs, you only care which nodes are connected - but you don’t care about distances, geometry, etc. It’s not defined there, and it isn’t needed.
Topological structure - You forget distances and angles. All that matters is what stays the same when you stretch or squash the space, as long as you don’t cut or glue.
Algebraic structure - You define operations (like +, ×) and rules for how things behave. Example of this is groups of symmetries.
Order structure - You define a way to say one element is “greater” or “before” another.
So, topology is an example of a mathematical structure that isn’t about measurements - angles, lengths, coordinates - and instead focuses on what stays the same when you stretch, squash, or bend things.
I will show you why such discipline is useful, and how we define such weird spaces that still have important characteristics, even without such basic things as lengths or angles.
How Topology Started
I like how the first-ever topological problem will tell you a lot about what topology is good for.
Back in the 18th century, people in the city of Königsberg (in Prussia) wondered: Is it possible to take a walk through town that crosses each of the city’s seven bridges exactly once?
At first glance, this seems like a geographic or navigation problem. But to solve it, Leonhard Euler looked at the problem in a topological way. He ignored distances, angles, and even the actual map of the city. He simplified the problem into what we now call a graph.
He turned each land mass into a vertex (a node on the graph), and each bridge into an edge (a “line” on the graph). What mattered wasn’t how long the bridges were or what shape the islands were - just how things were connected.
This was one of the first major moments in topology.
Euler showed that no such walk is possible in Königsberg - there’s no way to cross all seven bridges without repeating one. He also established mathematical rules that determine whether other problems like this have a solution, or not.
Notice that Euler focused on what stayed the same, even if you stretched or bent the map. That is, he focused on “topological invariants” (we will get to this later).
That’s the thing I like about topology - it helps us reason about problems in a more abstract, flexible way, and get rid of details that aren’t important.
Key Concepts in Mathematical Topology
Intuitively, we define topology as the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending.
Formally, we define a topological space via open sets (elements of a set we denote tau). Open sets are a way to talk about closeness and continuity without relying on distances or angles. They let us define what it means for a point to be “inside” a region, without having to measure how far it is from the boundary.
The set X with the tau is called a topological space, and elements of the tau are called open sets.
Homeomorphisms (maps between “topologically similar” spaces)
And since in math, we like to map spaces to other spaces, we also define a homeomorphism - a function between two spaces that preserves their topological structure.
So when people say “a donut is the same as a mug,” they don’t mean they’re literally equal. They mean there’s a homeomorphism between them. Topologically, they’re indistinguishable.
Now back to “topological invariants”…
Now that we have defined a topological structure, we can define some very fundamental properties of topological spaces, which will be very useful for proving things about them. We call such properties “invariants”.
A topological invariant is a property of a space that stays the same no matter how you continuously deform it. If you stretch, squash, or bend a shape.
Topological invariants help us classify spaces and prove when two things are not topologically equivalent.
Some examples of topological invariants:
Euler characteristic – a relation between the number of vertices, edges, and faces of spaces
Number of holes in the space (genus)
Connectedness – is the space one piece or not?
Compactness – is it "bounded and closed" in a loose sense?
Example 1: The Euler Characteristic
This is the first example of using topology. There is a relationship between number of vertices, edges, and faces of surfaces.
The number 2 here is called the Euler characteristic, and it’s one of the simplest topological invariants.
Here’s how it works. Take a cube: it has 8 vertices, 12 edges, and 6 faces. Plug it in:
8 - 12 + 6 = 2.
If you squash, stretch, or deform the cube, it doesn’t matter. As long as you don’t poke a hole or tear it apart, the Euler characteristic stays the same.
What changes the characteristics is adding holes. If you turn your shape into a donut (torus), the Euler characteristic becomes 0. But why? How do we even draw the vertices, edges, and faces on a torus?
To see why, we need to find a good way to denote the V, E, and F on the torus. It helps to flatten the torus into a square.
Notice that when you glue the square together into the torus, all the corners become the same vertex, each pair of opposite edges becomes one edge, and the whole square is a single face. So you get V = 1, E = 2, F = 1, and the Euler characteristic of donut is 1 - 2 + 1 = 0.
I don’t want to confuse you here, but this deformation of square into torus is NOT homeomorphism, because it’s not continuous deformation. You tear and glue the spaces here. Therefore, Euler characteristic of the square is also not the same as the torus. Euler characteristics of square is V - E + F = 4 - 4 + 1 = 1. We just used the square here to help us describe the torus.
Example 2: The Hairy Ball Theorem
This one informally says that you can’t comb a hairy ball flat without creating a cowlick.
Formally, the Hairy Ball Theorem states that any continuous tangent vector field on an even-dimensional sphere must have a point where the vector is zero.
In reality, this theorem says, for example, that if you're picturing the Earth, at every point you try to draw a wind vector that’s continuous across the surface, there must be a place with no wind. This isn’t a limitation of physics - it’s baked into the topology of the sphere.
Again, the result is topological - it doesn’t depend on how large the sphere is, or whether it's distorted - only on its topological properties, like compactness and genus (number of holes, which is 0 here).
More Real-World Examples
Here are just a few places where topology is an active tool in science and technology. I won’t go into details here, but I hope it really persuades you that topology isn’t just math for the sake of math.
Neuroscience: Researchers use persistent homology to study brain dynamics, especially how groups of neurons activate together over time. This reveals shapes in the data that aren’t visible through classical statistical methods:
Sensor networks: If you scatter sensors across a space (a forest, a building), topology helps determine whether the entire area is covered, or if there are holes in the coverage.
Materials science: Some materials have topological properties at the quantum level. Topological insulators, for example, conduct electricity on their surfaces while remaining insulating inside. This weird behavior isn’t explained by geometry or chemistry alone—it comes from the topology of their quantum states.
Biology: DNA often gets tangled, looped, or knotted. Enzymes that untangle DNA are modeled using ideas from knot theory—a subfield of topology.
Quantum computing: Some of the most noise-resistant forms of quantum computing rely on storing information in topological properties of particle braids, not in the particles themselves. You can twist or stretch the braids, but as long as you don’t break the braid (e.g., don’t change its topological properties), the information is preserved. I wrote more on this here:
Topology vs Other Disciplines
Why do we need topology if we already have geometry?
The difference is in what we care about. Geometry is about precise measurements -angles, lengths, and areas. Topology ignores those and focuses on structure and continuity. For example, topology doesn’t care if a circle is big or small—just that it’s “looped” with no endpoints.
What Does Topology Have To Do With Graph Theory?
You might have noticed that a lot of the pictures in this article were suspiciously close to graphs. We mentioned vertices, edges, .... And yes, topology and graph theory are close to each other and overlap by nature. It’s because both of these disciplines don’t care about the length and sizes of things that much; they just care about the shapes or how the things are connected.
Topology and graph theory both started with the Königsberg bridge problem, by abstracting land masses and bridges into vertices and edges. Both of these mathematical branches don’t care about lengths or angles, only about connections.
A graph is a set of elements that have some notion of connection (even family members).
Here are my other math blog posts, in case you like this one. Enjoy & leave me a comment if you liked this one. :)
Gödel's Incompleteness Theorems Explained Simply
This is my attempt to explain Gödel’s Incompleteness Theorems in a way that actually makes sense.
Excellent read. Thanks for sharing 😊