Topology

name: topology

description: > Expert topology assistant for mathematicians and students. Use this skill whenever the user needs: help with topological spaces, continuity, compactness, connectedness, homotopy theory, or homology. Includes both point-set topology and algebraic topology. trigger: Any mathematical problem involving topological concepts - from abstract theory to applied topology. license: MIT compatibility: opencode metadata: audience: mathematicians category: mathematics

Topology — Spaces, Continuity, and Algebraic Invariants

Covers: Point-Set Topology · Continuity · Compactness · Connectedness · Homotopy · Homology · Manifolds

Point-Set Topology

Topological Space

A set X with collection τ of subsets (open sets) satisfying:

1. ∅, X ∈ τ
2. Finite intersection ∈ τ
3. Arbitrary union ∈ τ

(X, τ) is a topological space.

Examples

Interior and Closure

• Interior: largest open set contained in A
• Closure: smallest closed set containing A

Int(A) = complement of closure of complement.

Boundary

∂A = closure(A) \ interior(A)

Points where A "meets" its complement.

Basis

Collection B such that:

1. For x ∈ B, x ∈ some B ∈ B
2. If x ∈ B₁ ∩ B₂, exists B₃ with x ∈ B₃ ⊂ B₁ ∩ B₂

Top generated by basis elements.

Subspace Topology

Inherited from ambient space:

U ⊂ Y is open in Y ⇔ ∃ O open in X with U = O ∩ Y

Product Topology

Product of topological spaces:

π_i: X₁ × X₂ → X_i

Open sets generated by products of opens.

Quotient Topology

quotient map q: X → Y:

U ⊂ Y is open ⇔ q^{-1}(U) open in X

Continuous Maps

f: X → Y is continuous if:

f^{-1}(open in Y) is open in X

Equivalent: preimages of closed sets are closed.

Homeomorphism

Bijection with continuous inverse. Spaces are topologically equivalent.

Topological Invariants

Properties preserved under homeomorphism:

• Connectedness
• Compactness
• Countability axioms
• Separation axioms

Separation Axioms

T₀ (Kolmogorov)

For any two distinct points, at least one has neighborhood not containing other.

T₁ (Frechet)

Singletons are closed.

T₂ (Hausdorff)

Any two points have disjoint neighborhoods.

Most "nice" spaces are Hausdorff.

Regular

Point and closed set can be separated by neighborhoods.

Normal

Two disjoint closed sets can be separated.

T₄ = normal + T₁.

Summary Table

Compactness

Definition

Every open cover has finite subcover.

Finite: compact automatically. Infinite: need property.

Heine-Borel (Rⁿ)

A set is compact ⇔ closed and bounded.

Not true in infinite dimensions!

Continuous Image

Continuous image of compact is compact.

Corollaries:

• Maximum/minimum exist for continuous functions on compact sets
• Continuous maps to Hausdorff are closed

Sequential Compactness

Every sequence has convergent subsequence. Equivalent to compactness in metric spaces.

Products (Tychonoff)

Arbitrary product of compact spaces is compact. Very powerful theorem.

Locally Compact

Each point has compact neighborhood. Rⁿ is locally compact. R (with lower limit topology) is not.

Paracompact

Every open cover has locally finite refinement. Manifolds are paracompact (partition of unity exists).

Covering Maps

p: E → B continuous, surjective:

• p^{-1}(U) disjoint union of opens in E
• Locally trivial

Fundamental in fiber bundle theory.

Connectedness

Definition

Cannot be partitioned into two nonempty disjoint opens. Equivalently: only clopen sets are ∅ and X.

Path Connected

Continuous path between any two points: γ: [0,1] → X with γ(0)=x₀, γ(1)=x₁.

Path connected ⇒ connected (not reverse).

Connected Components

Maximal connected subsets. Components are closed.

Locally Connected

Every point has basis of connected neighborhoods.

Simply Connected

• Path connected
• All loops contractible (π₁ = 0)

Fundamental Group

π₁(X, x₀) = homotopy classes of loops based at x₀:

• Group operation: concatenation
• Invariant under homeomorphism

Higher Homotopy Groups

π_n(X) = homotopy classes of maps S^n → X.

Homology Groups

H_n(X): algebraic topology invariant. For simplicial complexes: boundary operators.

Exact Sequences

... → H_n(A) → H_n(X) → H_n(X,A) → H_{n-1}(A) → ...

Powerful computational tool.

Metric Spaces

Metric

Function d: X × X → R satisfying:

1. d(x,y) ≥ 0, d(x,y)=0 ⇔ x=y
2. d(x,y) = d(y,x)
3. d(x,z) ≤ d(x,y) + d(y,z)

Metric Topology

Open balls generate topology:

B_r(x) = {y | d(x,y) < r}

Examples

• Euclidean: d(x,y) = |x-y|
• Manhattan: d(x,y) = Σ|x_i-y_i|
• Maximum: d(x,y) = max|x_i-y_i|
• Discrete: d(x,y) = 0 if x=y, else 1

Completeness

Cauchy sequence converges:

• R is complete
• Q is not

Baire Category

Countable union of nowhere dense sets has empty interior. Complete metric spaces are Baire.

Banach Spaces

Complete normed vector spaces.

Hilbert Spaces

Complete inner product spaces.

Fixed Point Theorems

Banach: Contraction on complete metric space has unique fixed point.

Brouwer: Continuous map of ball to itself has fixed point.

Schauder: Compact convex set to itself.

Norms

‖x‖ = √⟨x,x⟩:

• ‖x‖ ≥ 0, =0 ⇔ x=0
• ‖αx‖ = |α|‖x‖
• ‖x+y‖ ≤ ‖x‖ + ‖y‖

Homotopy

Homotopy of Maps

Maps f,g: X → Y are homotopic if: F: X × [0,1] → Y continuous F(x,0) = f(x), F(x,1) = g(x)

Write f ≃ g.

Homotopy Equivalence

X ≃ Y if there exist f: X → Y, g: Y → X with:

g ∘ f ≃ id_X, f ∘ g ≃ id_Y

Retracts, deformation retracts.

Contractible

Space homotopic to point. All homotopy groups trivial.

Homotopy Groups

π_n(X):

• n=0: # components
• n=1: fundamental group
• n≥2: abelian

Hurewicz Theorem

First non-zero homology → homotopy. Relates algebraic invariants.

Cellular Homotopy

CW complexes:

• n-skeleton
• Attach n-cells via attaching maps

CW Complexes

• Building blocks: cells
• Closure finite, weak topology
• Common in algebraic topology

Homotopy Extension

If A ⊂ X and f: A → Y extends to X, then any homotopy of f|_A extends.

Important for cofiber sequences.

Excision

For covering: H_n(X, A) ≈ H_n(X\B, A\B) for B ⊂ A interior.

Mayer-Vietoris

For X = U ∪ V: ... → H_n(U ∩ V) → H_n(U) ⊕ H_n(V) → H_n(X) → ...

Manifolds

Topological Manifold

Hausdorff, second-countable, locally Euclidean. Each point has neighborhood homeomorphic to R^n.

Smooth Manifold

Smooth atlas: transition maps C^∞. Differential topology.

Examples

Charts and Atlases

Chart: φ: U → R^n. Atlas: collection covering manifold.

Tangent Space

T_pM: derivations at p. Cotangent space: dual.

Smooth Maps

f: M → N is smooth if composed with charts.

Submanifolds

Embedded: locally like coordinate subspace. Immersed: injective derivative.

Orientability

Consistent orientation of tangent spaces. S^n orientable for all n. RP^n orientable only for n odd.

Morse Theory

Critical points of functions → topology. Gradient flows → cell structure.

Morse Inequalities

critical points of index k ≥ b_k.

De Rham Cohomology

Differential forms: H^k_{dR}(M) ≅ H_k(M; R)

Integration on Manifolds

Stokes' theorem:

∫_M dω = ∫_{∂M} ω

Generalizes fundamental theorem.

Algebraic Topology

Simplicial Complex

Vertices, edges, triangles, tetrahedra... Finite, combinatorial.

Simplicial Homology

Chain groups C_k:

• k-chains: formal sums of k-simplices
• Boundary operator ∂_k: k → k-1

Cycles: ker ∂k Boundaries: im ∂{k+1} H_k = ker/im

Singular Homology

More general: continuous maps of standard simplices.

Functoriality

Continuous maps → group homomorphisms. Homotopic maps → same homomorphism.

Euler Characteristic

χ = Σ (-1)^k b_k

For simplicial complex: vertices - edges + faces - ...

Homology Computations

• Exact sequences
• Universal coefficient theorem
• Künneth formula

Cohomology

Dual to homology:

• Cup product gives ring structure
• Poincaré duality

Cup Product

H^*(X;R) becomes graded ring. Cup product encodes topology.

Intersection Theory

Intersect submanifolds:

• Orientations matter
• Link to cohomology

Vector Bundles

Rank k fiber bundles:

• Tangent bundle
• Normal bundle

Characteristic Classes

• Stiefel-Whitney: mod 2
• Chern: complex
• Pontryagin: real

Fixed Point Theorems

Brouwer Fixed Point

Any continuous f: B^n → B^n has fixed point.

Applications: economics, game theory.

Schauder Fixed Point

Compact convex → continuous map has fixed.

Banach Fixed Point

Contraction mapping on complete metric space: unique fixed point, convergent iteration.

Lefschetz Fixed Point

For compact polyhedron: L(f) = Σ (-1)^k Tr(f_* : H_k(X) → H_k(X))

If L(f) ≠ 0, f has fixed point.

Index Theory

For manifolds, degree of map:

• Fixed points have index
• Sum = degree of map on boundary

Categorical Perspective

Categories

Objects: spaces, groups, ... Morphisms: continuous maps, homomorphisms

Functors

Map categories:

• Homology: Top → Ab
• Fundamental group: Top* → Grp

Natural Transformations

Between functors.

Limits and Colimits

• Product, pullback
• Coproduct, pushout

Adjunctions

(F ⊣ G):

• Hom(FX, Y) ≅ Hom(X, GY)

Example: free-forgetful for groups.

Homotopy Category

quotient by homotopy equivalence.

Applications

Data Analysis

• Persistent homology
• Topological data analysis
• Mapper algorithm

Robotics

• Configuration space topology
• Motion planning

Physics

• Topological insulators
• Quantum field theory
• Knot invariants

Knot Theory

• Alexander polynomial
• Jones polynomial
• Seifert surfaces

Topological Quantum Field Theory

• Path integrals
• Modular invariance

String Theory

• Calabi-Yau manifolds
• Mirror symmetry

Common Errors to Avoid

• Assuming compactness in infinite dimensions
• Confusing path-connected with connected
• Forgetting separation axioms
• Mixing up homology and homotopy groups
• Not checking continuity in definitions
• Confusing subspace with quotient topology
• Using wrong metric for problem
• Ignoring orientability in integration
• Assuming contractible implies trivial topology
• Not distinguishing local and global properties

Key References

• Topology by Munkres — Standard intro
• Algebraic Topology by Hatcher — Comprehensive
• Differential Topology by Guillemin & Pollack — Manifolds
• Elements of Algebraic Topology — Homology