Part VI: Geometric and Topological Structures

Topology studies properties preserved under continuous deformation — stretching, bending, twisting — but not tearing or gluing. Geometry adds measurement. Together they capture the shape of space.

6.1 Topological Spaces

Definition: A topological space is a set X with a collection τ of subsets (the open sets) satisfying:

1. ∅ and X are open
2. Arbitrary unions of open sets are open
3. Finite intersections of open sets are open

Basic notions:

• Closed set: complement of an open set
• Closure cl(A): smallest closed set containing A
• Interior int(A): largest open set contained in A
• Boundary ∂A = cl(A) ∖ int(A)
• Neighborhood of x: any open set containing x
• Limit point: x is a limit point of A if every neighborhood of x contains a point of A distinct from x
• Dense subset: A is dense in X if every non-empty open set meets A

Separation axioms (T-axioms):

• T₀: distinct points are topologically distinguishable
• T₁: all points are closed
• T₂ (Hausdorff): distinct points have disjoint neighborhoods — limits of sequences are unique
• T₃ (Regular): closed sets and points can be separated
• T₄ (Normal): closed sets can be separated

6.2 Metric Spaces

Definition: A metric space (X, d) is a set X with a distance function d: X×X → ℝ≥0 satisfying:

1. d(x, y) = 0 ↔ x = y (identity of indiscernibles)
2. d(x, y) = d(y, x) (symmetry)
3. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

Every metric space is a topological space: open balls B(x, r) = {y : d(x,y) < r} generate the topology.

Completeness: A metric space is complete if every Cauchy sequence converges. ℝ is complete; ℚ is not. The completion of ℚ with the standard metric is ℝ.

Compactness: A metric space is compact if every sequence has a convergent subsequence (sequential compactness, equivalent to topological compactness for metric spaces). Compact spaces are "finite-like" in many ways.

Key examples:

• ℝⁿ with Euclidean metric
• Function spaces C([0,1]) with sup metric — infinite-dimensional but complete
• Fractal sets — Cantor set, Sierpiński triangle — with inherited metrics

6.3 Continuity and Homeomorphism

Continuous map: f: X→Y is continuous if f⁻¹(U) is open in X whenever U is open in Y.

Homeomorphism: A bijective continuous map with a continuous inverse. Homeomorphic spaces are "topologically identical" — they have the same topological properties.

Topological properties: Properties preserved by homeomorphisms: connectedness, compactness, path-connectedness, dimension, Euler characteristic. These are the "topological invariants."

Connectedness: A space is connected if it cannot be partitioned into two disjoint non-empty open sets. A coffee cup and a donut are homeomorphic (both have one hole). A sphere and a torus are not.

6.4 Fundamental Group and Homotopy

Homotopy: Two continuous maps f, g: X→Y are homotopic if there is a continuous H: X×[0,1]→Y with H(x,0) = f(x) and H(x,1) = g(x). H deforms f into g continuously.

Fundamental group π₁(X, x₀): The group of homotopy classes of loops based at x₀ under concatenation.

• π₁(ℝⁿ, 0) = 0 (trivial — ℝⁿ is simply connected)
• π₁(S¹, 1) = ℤ (loops around the circle are counted by winding number)
• π₁(torus, x₀) = ℤ × ℤ (independent winding around each handle)

Higher homotopy groups π_n(X, x₀): Homotopy classes of maps from the n-sphere Sⁿ to X. These detect higher-dimensional "holes."

Key theorems:

• Brouwer Fixed Point Theorem: Every continuous map from the closed n-disk to itself has a fixed point
• Jordan Curve Theorem: A simple closed curve in ℝ² divides it into two regions
• Poincaré Conjecture (proved by Perelman, 2003): Every simply connected closed 3-manifold is homeomorphic to S³

6.5 Manifolds

Topological manifold: A Hausdorff, second-countable topological space that is locally homeomorphic to ℝⁿ. The integer n is the dimension.

Smooth manifold: A topological manifold with an atlas of charts whose transition maps are smooth (C∞).

Examples:

• ℝⁿ, open subsets of ℝⁿ
• The n-sphere Sⁿ = {x ∈ ℝⁿ⁺¹ : |x| = 1}
• The n-torus Tⁿ = S¹ × … × S¹
• Projective spaces ℝℙⁿ, ℂℙⁿ
• Lie groups (smooth manifolds with group structure)

Tangent space: At each point p of a smooth manifold M, the tangent space T_p M is a vector space of the same dimension as M, capturing the "directions" one can move from p.

Riemannian manifold: A smooth manifold with a Riemannian metric — a smooth family of inner products on each tangent space — allowing measurement of lengths, angles, and volumes.

6.6 Differential Geometry

Curvature: The Riemann curvature tensor R measures how much a vector rotates when parallel-transported around a loop. Curvature is zero iff the manifold is locally flat.

Gaussian curvature (surfaces): For a 2-dimensional surface embedded in ℝ³, the Gaussian curvature K at a point is the product of the principal curvatures.

• Sphere: K = 1/r² > 0 everywhere
• Plane: K = 0 everywhere
• Saddle surface: K < 0 at saddle points

Gauss-Bonnet Theorem: For a compact surface M, ∫∫_M K dA = 2πχ(M), where χ is the Euler characteristic. Geometry and topology are linked: the integral of curvature is a topological invariant.

Connections and parallel transport: A connection on a manifold specifies how to "parallel transport" vectors along curves. The curvature measures the holonomy — how much a vector rotates after transport around a closed loop.

Einstein's field equations: General relativity is differential geometry. The Einstein tensor G (a curvature measure) equals 8πG/c⁴ times the stress-energy tensor T. The geometry of spacetime is determined by the distribution of matter and energy.

6.7 Algebraic Topology

Chain complexes and homology: Assign to a topological space X a sequence of abelian groups H_n(X) (the homology groups) that count "n-dimensional holes":

• H₀(X): the free abelian group generated by connected components
• H₁(X): abelianization of π₁ — one-dimensional loops
• H₂(X): two-dimensional "enclosed volumes"

Cohomology: Dual theory; cohomology groups H^n(X) are the Hom duals of homology. De Rham cohomology identifies H^n with equivalence classes of differential n-forms, connecting topology to analysis.

Euler characteristic: χ(X) = ∑ (-1)ⁿ rank H_n(X). For a convex polyhedron: χ = V − E + F = 2.

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