A homeomorphism or continuous transformation, is an

• equivalence relation and
• one-to-one (injective) correspondence between points in two geometric figures or topological spaces that is continuous in both directions.
• preserves distances is called an isometry.
• Affine transformations

Those are the technical words. Here’s the truth behind them: topology is geometry with the ruler thrown away. It only cares what’s next to what, and whether you can get from here to there without tearing.

The core idea, in plain words

Most geometry is about measurement: lengths, angles, areas. Topology deliberately forgets all of that. Two shapes count as the same if one can be bent, stretched, squeezed or twisted into the other without tearing and without gluing. Such a deformation is the homeomorphism above: a one-to-one correspondence that is continuous both ways. Continuity means “nearby points go to nearby points”; continuity both ways means you never tear and never glue.

Formally, a topological space is a set together with a choice of which subsets are open, the “neighbourhoods”, such that:

1. the empty set and the whole set are open;
2. any union of open sets is open;
3. any finite intersection of open sets is open.

That’s all. No distances. Just a notion of “nearby”. From this thin soup grow continuity, connectedness, compactness and holes.

Notice the link to the Set Relations pages: “is homeomorphic to” is an equivalence relation, reflexive, symmetric and transitive. Topology chops the universe of shapes into equivalence classes and asks what’s left when you forget the details.

An everyday analogy: the coffee mug and the doughnut

The old joke: a topologist can’t tell a coffee mug from a doughnut. Imagine a mug made of soft clay. Squash the cup part down into the handle and smooth it out, and you get a ring, a doughnut. No tearing, no gluing. Both have exactly one hole, and that is the only thing topology sees. A ball, on the other hand, can never become a doughnut without punching a hole in it.

A tube map is the same idea in daily life. The real distances and curves are thrown away; only the connections, which station is next to which, survive. And that’s all you need to get across town.

A worked example: Euler’s formula

Draw any connected network on a sphere (or a flat sheet): $V$ dots, $E$ lines, $F$ regions (counting the outside). Then

Check a cube: $8 - 12 + 6 = 2$. ✔ A tetrahedron: $4 - 6 + 4 = 2$. ✔ A single dot on a sphere: $1 - 0 + 1 = 2$. ✔

Now draw on a doughnut. A square grid wrapped round it with 1 dot, 2 lines (one round the hole, one through it) and 1 square: $1 - 2 + 1 = 0$.

That number, the Euler characteristic, doesn’t depend on how you draw the network, only on the surface. Sphere: 2. Doughnut: 0. Two-holed doughnut: $-2$. In general a surface with $g$ holes has

So the sphere and the doughnut cannot be homeomorphic, because their characteristics differ. That’s topology’s main trick: find a number (or a Group) that can’t change under deformation, then use it to prove two shapes are different.

A picture in words: an ant on a rubber band

Picture an ant walking along a closed rubber band. Stretch the band into a square, a long oval, a wiggly loop: the ant’s journey is “the same”, it goes once round and comes back. Now cut the band. The ant’s world is now a line with two ends, and it can never return without turning back. Cutting is the one thing topology forbids, because it changes the answer to “can I go round?”

The loops the ant can walk, up to deformation, form the fundamental group. On a circle it’s the integers (go round $n$ times), the group $\mathbb{Z}$. On a sphere every loop can be shrunk to a point, so it’s trivial.

graph LR
  S["Sphere: χ = 2"] -.->|"not homeomorphic"| T["Doughnut: χ = 0"]
  M["Coffee mug: χ = 0"] ---|"homeomorphic"| T

Common misconceptions

• “Topology is about shapes looking alike.” Looks are irrelevant. A square and a circle are the same; a circle and a figure-eight are not.
• “Any continuous bijection is a homeomorphism.” The inverse must be continuous too. Wrap the half-open interval $[0, 1)$ round a circle: it’s continuous and one-to-one, but the inverse tears the circle open at one point.
• “Open sets are ‘sets without their edge’.” In $\mathbb{R}$ that’s the intuition, but in a general space “open” just means “declared to be a neighbourhood”, and strange spaces have strange open sets.
• “Topology is useless because it ignores measurement.” Ignoring measurement is its power: it finds facts robust to any amount of wobble. GPS errors, protein folding and data analysis all use it.

How it connects

• Down to Set Theory (a topology is a collection of sets) and Set Relations (homeomorphism is an equivalence relation).
• Sideways to Affine Transformations, which keep lines and ratios, and isometries, which keep distances. Topology keeps the least of all.
• Up to algebraic topology, where Groups measure holes, and to Category Theory, which was invented to organise exactly that.
• Out to Relativity, where the shape of the universe is a topological question, and to Dynamics, where Poincaré invented much of topology to study planetary orbits.

The truth revealed: some facts survive any amount of stretching. Topology is the hunt for them.