Topology

Topology studies continuity, open sets, compactness, connectedness, and invariants preserved by homeomorphism.

Knot theory is the largest field, with nearly half of the subject's articles. It starts from knots, links and ambient isotopy and knot diagrams, and goes on to geometric questions: ropelength and crossing number, the distortion of knots and the quadrisecant theorem, with slice knots on the four-dimensional side.

General topology covers dimension theory and the limits and products that build new spaces, and descriptive set theory characterizes Baire space, Cantor space and the rationals and studies the Banach–Mazur game. Algebraic topology includes the Hurewicz theorem in degree one and the homotopy groups of spheres, and manifolds and geometric topology runs from the classification of surfaces to the Poincaré conjecture in dimensions three, four and at least five and the annulus theorem in dimensions at least four. The Hilbert cube and absolute retracts belong to continuum theory and infinite-dimensional topology.

• General topology
• Descriptive set theory
• Continuum theory and infinite-dimensional topology
• Topological groups
• Algebraic topology
• Manifolds and geometric topology
• Knot theory

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