Short answers, pulled from the story.
Topology is 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. It studies properties that survive deformation without closing holes, opening holes, tearing, gluing, or passing an object through itself.
A coffee mug and a doughnut are homeomorphic, meaning one can be deformed into the other without cutting or gluing. A pliable torus shaped like a doughnut can be reshaped into a coffee mug by creating a dimple and enlarging it while shrinking the central hole into the mug's handle. This idea is sometimes called the Topologist's Breakfast.
Johann Benedict Listing introduced the term Topologie in his work Vorstudien zur Topologie, written in German in 1847. He had used the word for ten years in correspondence before it appeared in print, and the English form topology was used in 1883 in Listing's obituary in the journal Nature.
The Seven Bridges of Konigsberg problem asked whether a route could cross each of the town's seven bridges exactly once. Leonhard Euler proved it impossible in his 1736 paper, showing the result depended only on connectivity rather than distances, which led to the branch of mathematics known as graph theory.
The main branches are general topology, which is also called point-set topology and provides the set-theoretic foundations, algebraic topology, which uses algebraic invariants such as homotopy groups, homology, and cohomology, differential topology, which studies differentiable functions on differentiable manifolds, and geometric topology, which focuses on low-dimensional manifolds of dimensions 2, 3, and 4.
In biology, knot theory is used to study how enzymes cut, twist, and reconnect DNA, and circuit topology classifies folded proteins and nucleic acids. In physics, topology applies to condensed matter, quantum field theory, quantum computing, and cosmology, and David Thouless, Duncan Haldane, and Michael Kosterlitz won the 2016 Nobel Prize in Physics for their work on topological orders.