Rubber-sheet geometry
Topology asks: what properties of a shape don't change when you stretch, bend, or twist it (without tearing or gluing)? A coffee cup and a donut are topologically equivalent — you can continuously deform one into the other. A sphere and a donut are not, because you'd need to make a hole.
The Euler characteristic
The first topological invariant — a quantity that stays the same under continuous deformation — was discovered by Euler in 1752. Euler's formula V − E + F = 2 holds for any convex polyhedron because all such surfaces are topologically spheres. Change the topology (add a hole, make a torus) and the number changes.
Key ideas
- Continuous maps: functions that don't tear things apart.
- Homeomorphism: a continuous map with a continuous inverse — the topological notion of "sameness."
- Connectedness: can you get from any point to any other without leaving the space?
- Compactness: a topological version of "closed and bounded."
- Genus: the number of holes in a surface.
The Poincaré conjecture
Topology's most famous modern achievement is the proof of the Poincaré conjecture: every simply-connected, closed 3-manifold is topologically a 3-sphere. Posed in 1904 by Henri Poincaré, it was solved in 2002–2003 by Grigori Perelman using Ricci flow — the first of the seven Millennium Prize Problems to be resolved.
Why topology matters
Topology is everywhere in modern mathematics and science:
- Physics: topological phases of matter, topological insulators.
- Biology: the topology of DNA knots.
- Data science: topological data analysis (TDA) for finding structure in high-dimensional data.
- Cosmology: the topology of the universe itself is an open question.