Topology versus geometry
Geometry measures distances and angles; topology tracks properties unchanged by continuous deformation.
$$X\simeq Y\quad\text{(homeomorphism)}$$
What this means
Geometry measures distances and angles; topology tracks properties unchanged by continuous deformation.
Deep dive: mathematical and physical mechanism
A homeomorphism is a continuous bijection with continuous inverse. A circle and a square are homeomorphic, while a circle and a line segment are not. Quantum topology typically concerns spaces of states, momentum-space eigenvectors, or order parameters—not the physical shape of a device.
How it appears in quantum phenomena
A band insulator may have a nontrivial topology even on a geometrically ordinary crystal.
Assumptions, caveats and limitations
Topological equivalence is not an assertion that all physical observables agree.
Connect the ideas
For context, see the Foundations of Topology topic guide and the reading references.