Open sets, continuity and compactness
Topology supplies a precise definition of continuity without first choosing a distance.
$$f^{-1}(U)\ \text{is open for every open }U$$
What this means
Topology supplies a precise definition of continuity without first choosing a distance.
Deep dive: mathematical and physical mechanism
A topology is a collection of open subsets closed under arbitrary unions and finite intersections. Compactness permits finite subcovers and helps make global band invariants well defined on the compact Brillouin torus.
How it appears in quantum phenomena
Periodic crystal momentum forms a compact torus in ideal crystalline models.
Assumptions, caveats and limitations
Compactness and spectral gaps are distinct requirements.
Connect the ideas
For context, see the Foundations of Topology topic guide and the reading references.