Stretch, bend, twist — but never cut, never glue. Under such deformations the cup with one handle and the torus with one hole are indistinguishable. Both are surfaces of genus one.
Geometry asks how big? Topology asks what shape, fundamentally? — counting holes, sides, edges, components, and other features that survive the squishing.
Citizens asked: can you walk through town and cross every bridge exactly once? Euler showed it was impossible — and in doing so, he ignored distance entirely. Only the connectivity of land masses mattered.
An Eulerian path exists iff at most two vertices have odd degree.
The first theorem of what would become graph theory and topology.
A homeomorphism is a continuous map with a continuous inverse. Two spaces are homeomorphic if such a map exists between them.
A circle and a square are homeomorphic. A circle and a figure-eight are not.
The open sets must contain the empty set and the whole space, and be closed under arbitrary unions and finite intersections. From this minimal data — no distance, no angle — the entire structure of continuity follows.
A function f : X → Y is continuous iff the preimage of every open set is open.
This is topology's great move: detach continuity from the real numbers and turn it into pure structure.
For any convex polyhedron — cube, tetrahedron, dodecahedron — vertices minus edges plus faces always equals two. The number doesn't care about the shape's specifics; it sees only the topology of the sphere.
For a torus, χ = 0. For a double torus, χ = −2. The invariant counts holes.
Take a strip of paper, give it a half-twist, glue the ends. Trace your finger along the surface — you return to the start having traversed both apparent sides. There is only one.
The simplest non-orientable surface. Cut it down the middle and a single longer loop with two twists results.
Glue two Möbius strips along their single edges and you get a closed non-orientable surface. It cannot be embedded in three-dimensional space without self-intersection — it lives properly in four dimensions.
Outside is inside. The bottle has no boundary, yet contains no enclosed volume.
Two knots are equivalent if one can be deformed into the other without cutting the string. The unknot, the trefoil, the figure-eight — each is genuinely distinct. Distinguishing them required new invariants: the Alexander polynomial (1928), the Jones polynomial (1984).
Knot theory now reaches into DNA recombination, quantum field theory, and statistical mechanics.
To each space attach an algebraic gadget — a group, a ring, a sequence — that depends only on the topology. If the gadgets differ, the spaces are not homeomorphic.
π₁(circle) = ℤ. The integer counts how many times a loop winds.
Every simply-connected, closed 3-manifold is homeomorphic to the 3-sphere.
Stated by Henri Poincaré, the question hung open for nearly a century. Grigori Perelman posted three preprints to arXiv in 2002–03 proving it via Hamilton's Ricci flow with surgery — and resolved Thurston's vast geometrization conjecture along the way.
He was awarded the Fields Medal (2006) and the $1M Clay Millennium Prize (2010). He declined both. "I'm not interested in money or fame."
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