Point-set and algebraic topology: open/closed sets, continuous maps, homeomorphism, compactness, connectedness, metric spaces, and the first rungs of algebraic topology (fundamental group, homotopy, homology). Existing nodes topology-space…
topology
Topological space
A set X equipped with a topology τ — a family of subsets closed under arbitrary union and finite intersection, including ∅ and X.…
Differential geometry (manifold)
A smooth manifold is a topological space locally homeomorphic to ℝⁿ with smoothly overlapping charts. Substrate for curvature, connection,…
Open set
A member of the topology τ on X. Open sets are closed under arbitrary union and finite intersection, and include ∅ and X.
Closed set
Complement of an open set. Closed sets are closed under arbitrary intersection and finite union.
Continuous map
A map f: X → Y is continuous iff f⁻¹(U) is open in X for every open U ⊆ Y. Generalises the ε-δ definition from analysis.
Homeomorphism
A bijective continuous map with continuous inverse. The equivalence relation of topology: 'the donut and the coffee cup'.
Metric space (X, d)
A set X with distance d: X×X → ℝ≥0 satisfying identity, symmetry, and triangle inequality. Every metric induces a topology; not every…
Compactness
A space is compact if every open cover has a finite subcover. Heine-Borel: in ℝⁿ, compact ⇔ closed and bounded. Extreme-value theorem:…
Connectedness
A space is connected if it is not a disjoint union of two nonempty open sets. Intermediate-value theorem is the prototype.
Hausdorff (T₂) space
A space where any two distinct points have disjoint open neighbourhoods. Limits of convergent sequences are unique.
Fundamental group π₁(X, x₀)
The group of homotopy classes of loops at a basepoint x₀ under concatenation. Captures the '1-dimensional holes' of X. π₁(S¹) = ℤ.
Homotopy
A continuous deformation between two maps f, g: X → Y, i.e. a map H: X × [0,1] → Y with H(·,0) = f and H(·,1) = g. Basis of homotopy…
Homology H_n(X)
A sequence of abelian groups assigned functorially to a space, detecting n-dimensional holes. Computed from chain complexes;…
Brouwer fixed-point theorem
Every continuous map f: Dⁿ → Dⁿ (closed n-disk) has a fixed point. Proved via homology or degree theory; foundational for game theory…
Separation axioms (T0–T4)
A hierarchy of regularity conditions on a topological space X: T0 (distinguishability), T1 (points closed), T2 (Hausdorff), T3 (regular +…
Tychonoff's theorem
The product of any family of compact topological spaces is compact (in the product topology). Equivalent to AC over ZF.
Urysohn's lemma
In a normal (T4) space, disjoint closed sets A, B admit a continuous function f : X → [0,1] with f≡0 on A and f≡1 on B. Root of…
CW complex
A space built inductively by attaching n-cells D^n via attaching maps S^{n−1} → X^{(n−1)}; 'closure-finite, weak-topology'. Combinatorial…
Simplicial complex
A set K of simplices closed under taking faces and intersections. Realises as a topological space |K|; basis for simplicial homology and…
Homotopy group π_n(X)
The set of basepoint-preserving homotopy classes of maps (S^n, *) → (X, x_0), with a group structure for n ≥ 1 (abelian for n ≥ 2). π_1 is…
Cohomology H^n(X)
The cohomology of a chain complex C^*: H^n = ker d^n / im d^{n−1}. Dual to homology; carries a cup-product ring structure and often better…
Covering space
A continuous surjection p : E → B such that every b ∈ B has a neighbourhood U with p^{−1}(U) a disjoint union of copies of U each mapped…
Fiber bundle
A map p : E → B with a typical fibre F and a local-triviality condition: every b has a neighbourhood U with p^{−1}(U) ≅ U × F over U. …
Singular homology
H_n(X; R) = H_n(C_*(X; R)), where C_n(X; R) is the free R-module on continuous maps Δ^n → X. Functorial, homotopy-invariant, and agrees…
de Rham cohomology
H^n_{dR}(M) = ker(d : Ω^n → Ω^{n+1}) / im(d : Ω^{n−1} → Ω^n) on a smooth manifold M. de Rham's theorem: isomorphic to singular cohomology…
Exact sequence
A sequence of modules/groups and maps A → B → C where im(A→B) = ker(B→C). Long exact sequences (homology pair, Mayer–Vietoris) and short…
Spectral sequence
Sequence of pages Eᵣ converging to graded object; computes (co)homology via successive approximation. Serre (fibration), Adams (stable…
Characteristic classes
Natural cohomology classes of vector bundles. Chern (complex), Pontryagin (real), Stiefel-Whitney (ℝ, mod 2). Obstruct trivialisations;…
Simplicial homology
Chain complex C_n = ℤ[n-simplices], ∂ alternating-face boundary. H_n = ker ∂_n / im ∂_{n+1}. Independent of triangulation.
Mayer-Vietoris sequence
Long exact sequence … → H_n(A∩B) → H_n(A)⊕H_n(B) → H_n(X) → H_{n−1}(A∩B) → … for X = A∪B. Computational workhorse.
Poincaré duality
Closed orientable n-manifold M: H_k(M) ≅ H^{n-k}(M) via cap with fundamental class. Compactly-supported version for open; Lefschetz for…
LES of fibration
Fibration F→E→B: … → πₙ(F) → πₙ(E) → πₙ(B) → π_{n−1}(F) → … Serre spectral sequence computes H*(E) from H*(B), H*(F).
Covering space theory
p: X̃ → X with evenly-covered neighborhoods. Galois correspondence: connected covers ↔ subgroups of π_1(X). Universal cover simply…
Seifert-Van Kampen theorem
X = U ∪ V path-connected open, U∩V path-connected: π_1(X) = π_1(U) *_{π_1(U∩V)} π_1(V). Computes fundamental groups via gluing.
Morse theory
Morse function f: M→ℝ with non-degenerate critical pts. Level-set topology changes at critical values; M built from cells of dim = Morse…
CW approximation
Every topological space is weakly equivalent to a CW complex; map f: X→Y is a weak equivalence iff induces iso on all πₙ. Whitehead.
Obstruction theory
Extending map over (n+1)-cells obstructed by class in H^{n+1}(X; π_n(Y)). Computes existence and classification of lifts in fibrations.
Bott periodicity
π_{n+2}(BU) ≅ π_n(BU) (complex), π_{n+8}(BO) ≅ π_n(BO) (real). Foundation of K-theory.
Cobordism
Oriented cobordism classes Ω^SO_* forms graded ring under disjoint union. Thom: Ω^SO_n = π_n(MSO). Exotic spheres via Kervaire-Milnor.
h-cobordism theorem
Simply-connected manifold pair (W; M, M') of dim ≥ 6 with inclusions htpy eqs is diffeomorphic to M × [0,1]. Smale 1961 (Fields); Poincaré…
Surgery theory
Systematic modification of manifolds by replacing Sᵏ×Dⁿ⁻ᵏ with Dᵏ⁺¹×Sⁿ⁻ᵏ⁻¹. Classifies manifolds up to h-cobordism;…
Borsuk-Ulam theorem
For every continuous map f: Sⁿ → Rⁿ, there exists a point x ∈ Sⁿ with f(x) = f(−x). Equivalently, no antipode-preserving continuous map Sⁿ…
Hairy ball theorem
No continuous nonvanishing tangent vector field can exist on an even-dimensional sphere S^(2n). Follows from Poincaré-Hopf: the sum of…
Jordan curve theorem
Every simple closed curve γ in the plane divides R² ∖ γ into exactly two connected components — a bounded interior and an unbounded…
Tietze extension theorem
A continuous real-valued function on a closed subspace A of a normal space X extends to a continuous function on all of X, preserving the…
Banach fixed-point theorem
A contraction T on a complete metric space has a unique fixed point x*, and the iteration x_{n+1} = T x_n converges to x* geometrically…
Baire category theorem
A countable intersection of open dense sets in a complete metric space (or locally compact Hausdorff space) is dense; equivalently, such a…
Invariance of domain
A continuous injective map f from an open set U ⊂ Rⁿ to Rⁿ has open image f(U) and is a homeomorphism onto its image. Brouwer 1912. …
Hurewicz theorem
For an (n−1)-connected space X (n ≥ 2), the Hurewicz map h_n: π_n(X) → H_n(X; Z) is an isomorphism. The n = 1 case gives π₁(X)^(ab) ≅…