Plain-language explanation.
Topology is the study of properties of spaces that are preserved under continuous deformations — stretching, bending, twisting — but not tearing or gluing. It is often described as 'rubber-sheet geometry': a coffee cup and a donut are topologically equivalent (both have one hole).
Core concepts and standard treatment.
Core topology starts from metric spaces (distance, open and closed balls, convergence, continuity — the epsilon-delta definition generalised), then moves to topological spaces (open sets, the topology axioms, examples — discrete, indiscrete, Euclidean, Zariski), continuous maps, homeomorphisms (topological equivalence), and fundamental topological properties: compactness (Heine-Borel theorem — closed + bounded in ℝⁿ), connectedness (path-connectedness, intermediate value theorem), and separation axioms (Hausdorff, T1, normal spaces).
Deeper theory, debates and edge cases.
Advanced topology covers algebraic topology (fundamental group π₁ — van Kampen's theorem; covering spaces; homology and cohomology — chain complexes, Betti numbers, Euler characteristic, Mayer-Vietoris sequence), differential topology (smooth manifolds, tangent bundles, Morse theory, de Rham cohomology — Poincaré duality), knot theory (knot invariants — Jones polynomial, Alexander polynomial; knot groups), and applied topology / TDA (topological data analysis — persistent homology, Vietoris-Rips complex, mapper algorithm — Carlsson).
How it is applied in practice.
At the research mathematician and data scientist applying TDA level, topology enables: persistent homology for data shape analysis (Ripser, Gudhi, Giotto-TDA — feature extraction from high-dimensional point clouds); knot theory applications in DNA topology (linking number, writhe — topoisomerases); quantum field theory (topological quantum field theories — Witten, Atiyah; topological insulators — Berry phase, Chern number); robotics configuration space analysis (motion planning, obstacle avoidance — topological complexity); and materials science (topological phases of matter — Nobel Prize 2016 — Thouless, Haldane, Kosterlitz).