Topology

Introduction

Topology is the study of properties preserved under continuous deformations. It’s sometimes called “rubber sheet geometry”—properties like connectedness and holes matter, but exact distances and angles don’t.

The field emerged in the 18th century with Euler’s solution to the Königsberg bridge problem and has since become fundamental to modern mathematics, with applications ranging from pure mathematics to data science and physics.

Topological Spaces

Definition

A topology on a set X is a collection τ of subsets (called open sets) satisfying:

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

The pair (X, τ) is called a topological space.

Examples

• Discrete topology: Every subset is open
• Indiscrete topology: Only ∅ and X are open
• Standard topology on ℝ: Open intervals generate it
• Metric topology: Open balls generate it
• Cofinite topology: Open sets are those whose complements are finite

Bases

A base ℬ generates a topology if every open set is a union of base elements.

For ℬ to be a valid base:

• ℬ covers X
• If B₁, B₂ ∈ ℬ and x ∈ B₁ ∩ B₂, there exists B₃ ∈ ℬ with x ∈ B₃ ⊆ B₁ ∩ B₂

Subbases

A subbase generates a topology by taking all finite intersections (forming a base) and then all unions.

Closed Sets

A set is closed if its complement is open.

• Closed sets are closed under finite unions and arbitrary intersections
• The closure of A is the smallest closed set containing A
• The interior of A is the largest open set contained in A

Continuity

Topological Definition

f: X → Y is continuous if the preimage of every open set is open.

• Equivalent: preimage of every closed set is closed
• Equivalent: f(closure of A) ⊆ closure of f(A)

This generalises the ε-δ definition from analysis.

Homeomorphisms

A bijection f: X → Y where both f and f⁻¹ are continuous.

• Topologically equivalent spaces
• Example: Coffee cup ≈ doughnut (both have one hole)
• Example: Open interval (0,1) ≈ ℝ

Topological Invariants

Properties preserved by homeomorphisms:

• Compactness
• Connectedness
• Number of holes
• Euler characteristic
• Fundamental group

Important Properties

Compactness

Every open cover has a finite subcover.

In ℝⁿ (Heine-Borel): Compact ⟺ closed and bounded

Properties:

• Continuous image of compact is compact
• Compact subset of Hausdorff space is closed
• Closed subset of compact space is compact
• Product of compact spaces is compact (Tychonoff)
• Continuous real-valued function on compact space attains max/min

Connectedness

A space is connected if it cannot be split into two disjoint non-empty open sets.

Path Connectedness: Any two points can be joined by a continuous path.

• Path connected ⟹ connected
• Connected ⟹ path connected (in “nice” spaces like manifolds)

Components: Maximal connected subspaces partition the space.

Examples:

• ℝ is connected
• ℚ is totally disconnected
• S¹ (circle) is connected but removing one point makes it disconnected

Separation Axioms

Axiom	Condition
T₀	Points are topologically distinguishable
T₁	Points are separated by open sets
T₂ (Hausdorff)	Distinct points have disjoint neighbourhoods
T₃ (Regular)	Points and closed sets can be separated
T₄ (Normal)	Disjoint closed sets can be separated

Most spaces encountered in analysis are Hausdorff. Metric spaces are always T₄.

Metric Spaces

Definition

A metric d: X × X → ℝ satisfying:

1. d(x, y) ≥ 0; d(x, y) = 0 ⟺ x = y (positive definiteness)
2. d(x, y) = d(y, x) (symmetry)
3. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

Examples

• Euclidean metric: d(x, y) = √(Σ(xᵢ - yᵢ)²)
• Discrete metric: d(x, y) = 1 if x ≠ y, else 0
• Taxicab metric: d(x, y) = Σ|xᵢ - yᵢ|
• p-adic metric: Used in number theory

Metric Topology

Open balls B(x, r) = {y : d(x, y) < r} form a basis.

Every metric space is:

• Hausdorff
• First countable
• Paracompact

Completeness

A metric space is complete if every Cauchy sequence converges.

• ℝ and ℂ are complete
• ℚ is not complete
• Every metric space has a completion

Quotient Spaces

Definition

Identify points via an equivalence relation . The quotient space X/ has equivalence classes as points.

Examples:

• Circle S¹ = [0,1] with 0 ~ 1
• Torus = square with opposite edges identified (same orientation)
• Klein bottle = square with opposite edges identified (one reversed)
• Möbius strip = rectangle with one pair of edges identified with a twist
• Real projective plane ℝP² = sphere with antipodal points identified

Quotient Topology

Finest topology making the quotient map continuous.

A set U in X/~ is open ⟺ its preimage in X is open.

Surfaces

Classification of Surfaces

Every closed surface is homeomorphic to one of:

• Sphere with g handles (orientable, genus g)
• Sphere with k cross-caps (non-orientable)

Orientable surfaces: Sphere, torus, double torus, …
Non-orientable surfaces: Projective plane, Klein bottle, …

Euler Characteristic

χ = V - E + F (for polyhedral surfaces)

Surface	χ
Sphere	2
Torus	0
Double torus	-2
Klein bottle	0
Projective plane	1

For orientable surfaces: χ = 2 - 2g (where g is the genus)

Algebraic Topology

Fundamental Group

π₁(X, x₀) — loops based at x₀, up to homotopy.

• Measures “1-dimensional holes”
• π₁(S¹) ≅ ℤ
• π₁(sphere) is trivial
• π₁(torus) ≅ ℤ × ℤ
• π₁(figure-eight) ≅ free group on two generators

Covering Spaces

A covering map p: E → B is a local homeomorphism where each point has an evenly covered neighbourhood.

• Universal cover: simply connected covering space
• Deck transformations relate to the fundamental group

Homology

Sequence of abelian groups Hₙ(X) measuring n-dimensional holes.

• H₀: connected components
• H₁: 1-dimensional holes (loops)
• H₂: 2-dimensional holes (voids)

Homology is often easier to compute than homotopy groups.

Homotopy

Continuous deformation of maps.

• Two maps are homotopic if one can be continuously deformed into the other
• Homotopy equivalence is weaker than homeomorphism
• Contractible spaces are homotopy equivalent to a point

Product and Subspace Topologies

Product Topology

For spaces X and Y, the product topology on X × Y has basis {U × V : U open in X, V open in Y}.

Properties:

• Projections are continuous
• Continuous maps into products ⟺ component maps are continuous

Subspace Topology

For A ⊆ X, the subspace topology has open sets {U ∩ A : U open in X}.

Applications

• Data analysis: Topological data analysis, persistent homology for shape recognition
• Physics: Topological phases of matter, topological insulators, quantum field theory
• Robotics: Configuration spaces for motion planning
• Computer graphics: Mesh topology, surface reconstruction
• Network analysis: Studying connectivity and holes in networks
• Biology: Protein folding, neural connectivity

Glossary

Term	Definition
Open set	Element of the topology
Closed set	Complement of an open set
Neighbourhood	Open set containing a point
Homeomorphism	Topological equivalence (bicontinuous bijection)
Genus	Number of handles on an orientable surface
Homotopy	Continuous deformation of maps
Manifold	Space locally homeomorphic to ℝⁿ
Simplex	Generalisation of triangle to n dimensions

Learning Resources

Books

• Topology by James Munkres — standard undergraduate/graduate textbook
• Introduction to Topology by Bert Mendelson — accessible introduction
• Experiments in Topology by Stephen Barr — intuitive, hands-on approach
• Algebraic Topology by Allen Hatcher — comprehensive, free online
• Counterexamples in Topology by Steen and Seebach — essential reference

Online Resources

• Topology videos by PBS Infinite Series
• Numberphile topology videos
• 3Blue1Brown — Essence of Topology

See Also

• Real Analysis — foundational concepts of continuity and limits
• Abstract Algebra — group theory for algebraic topology
• Discrete Mathematics — combinatorial aspects