Introduction to Topology

Nearness without distance, and the invariants that survive stretching

About this course

Point-set topology from open sets, bases and continuity through subspaces, products and quotients, connectedness and compactness in their several forms, the separation axioms, and a first course in homotopy: the fundamental group, winding number and the fixed point theorem.

What you will be able to do

  • Work with topologies through open sets, bases, closure, interior and boundary
  • Recognise continuity and homeomorphism, and build new spaces by subspace, product and quotient
  • Use connectedness and path-connectedness to tell spaces apart
  • Prove and apply compactness in its open-cover, Euclidean and metric forms
  • Compute the fundamental group of the circle and derive the fixed point theorem from it

What this course covers

  • Metric axioms and metric spaces
  • Verifying the triangle inequality for Euclidean, taxicab, and maximum metrics
  • Open balls and their shapes under different metrics
  • Definition of open sets and the proof that every open ball is open
  • Open set operations: arbitrary unions and finite intersections
  • Constructing witness radii: punctured plane, closed disk, and exterior of a closed ball
  • Subspace (relative) neighborhoods
  • Metric equivalence: comparing Euclidean, taxicab, and maximum metrics
  • Discrete metric and the bounded metric; all subsets open
  • Homotopy: definition and basic examples
  • Homotopy as an equivalence relation and composition lemma
  • Path components as a homotopy invariant
  • Homotopy equivalence and contractibility
  • Retractions and deformation retractions
  • Maps into spheres: antipodal lemma and extension theorem
  • Definition and equivalent forms of connectedness
  • Connectedness of the real line and the least upper bound property
  • Continuous images of connected spaces are connected
  • Intermediate value theorem and fixed-point theorem
  • Building connected spaces and distinguishing spaces by cut points
  • Convergence of sequences in topological spaces
  • Convergence in metric spaces and worked examples
  • Multiple limits and non-Hausdorff topologies
  • Hausdorff spaces and uniqueness of sequence limits
  • Local neighborhood bases and first countability
  • Nested countable local basis construction
  • Closure characterization via sequences; first-countability role
  • Sequential closedness vs. closedness; co-countable counterexample
  • Co-countable space is not first countable
  • Sequential continuity and its equivalence under first countability
  • Second countability, separability, and their relationships
  • Closing synthesis of countability and sequence tests
  • Closed subsets, continuous images, and compactness on the real line
  • Finite intersection property, nested closed sets, and limit points
  • Basis reduction, Lindelöf property, and accumulation of sequences
  • Definition of compactness and open covers
  • Compact sets in specific topologies and the subspace lemma
  • Cartesian product and projections
  • Product topology via open rectangles
  • Product topology equals Euclidean topology on ℝ²
  • Product topology as coarsest topology with continuous projections
  • Universal property: continuity coordinate by coordinate
  • Topological properties preserved by products
  • Infinite products: product topology vs. box topology
  • Path homotopy relative to endpoints
  • Path homotopy as an equivalence relation
  • Concatenation of paths
  • Reparameterisation lemma
  • The fundamental group
  • Induced homomorphisms and functoriality
  • Loops on the circle and the winding-number homomorphism
  • Motivation: open sets without metric distances
  • Definition of a topology and topological space
  • Discrete and indiscrete topologies
  • Verifying and repairing proposed topologies
  • Comparing topologies: finer and coarser
  • Intersection and join of topologies
  • Basis for a topology: definition and local characterization
  • Basis theorem: coverage and overlap refinement
  • Basis examples: metric balls, rectangles, rational intervals, lower-limit topology
  • Cofinite topology and non-metrizable spaces
  • Minimal neighborhoods on finite topological spaces
  • Nested Closed Intervals Lemma
  • Compactness of the Closed Interval
  • Compact Subsets of Euclidean Space Are Bounded and Closed
  • Heine-Borel Theorem
  • Extreme Value Theorem
  • Distance, Nearest Points, and Uniform Continuity on Compact Sets
  • Applications: Diameter, Cantor Set, and Topological Invariance
  • Paths and path connectedness
  • The topologist's sine curve
  • Components and local connectedness
  • Local path connectedness and topological invariants
  • Quotient topology construction
  • Quotient topology as finest topology; duality with products
  • Saturated sets and the universal property of quotient maps
  • Recognition theorem: quotient homeomorphism
  • R mod Z and maps out of the circle
  • Gluing the square: cylinder, torus, and Möbius band
  • Quotient maps can destroy topological properties
  • Interior, closure, and boundary via neighborhood questions
  • Closed sets: definition and basic examples
  • Closed set laws: arbitrary intersections and finite unions
  • Interior of a set: definition, point-wise criterion, and properties
  • Interior and finite intersections: exact identity and failure for infinite intersections
  • Closure of a set: definition, point-wise criterion, and properties
  • Closure laws: finite unions exact, intersections and infinite unions only inclusions
  • Complement duality between interior and closure
  • Boundary: definition, partition of X, and characterization via clopin sets
  • Dense subsets: definition, neighborhood criterion, and examples
  • Hausdorff spaces and separation axioms
  • Separating a point from a compact set in a Hausdorff space
  • Compact subsets of Hausdorff spaces are closed
  • Closed maps and continuous maps from compact to Hausdorff spaces
  • The tube lemma and compactness of products
  • Closed-graph theorem and agreement sets
  • One-point compactification and homeomorphism with the circle
  • The helix, even covering, and path lifting setup
  • Uniqueness of lifts on connected spaces
  • Lebesgue number and the path-lifting theorem
  • Homotopy lifting and the winding number
  • Fundamental group of the circle is the integers
  • Winding number about a point and degree of circle maps
  • Fixed-Point Property on the Interval as Precursor
  • No-Retraction Theorem for the Disc
  • Brouwer's Fixed-Point Theorem for the Disc
  • Constructing and Verifying the Exit-Point Map
  • Accumulation points yield convergent subsequences
  • Bolzano–Weierstrass theorem
  • Cauchy sequences and completeness
  • Total boundedness
  • Three-way equivalence: compact, sequentially compact, complete and totally bounded
  • Uniform continuity, extensions, and the contraction principle
  • Definition and verification of the subspace topology
  • Point-wise (local) criterion for relative openness
  • Relative open and closed sets — examples and pitfalls
  • Metric restriction and basis restriction
  • Exact closure formula for subspaces
  • Interior and boundary comparisons for subspaces
  • Transitivity of subspace construction and relative density
  • The separation axioms ladder (T0–T4)
  • T1 characterisation and corollaries
  • Regularity: characterisation, metric spaces, and products
  • Normality: characterisation, metric spaces, and the shrinking lemma
  • Regular Lindelöf spaces are normal; the lower limit plane fails normality
  • Urysohn's lemma and the metrization theorem
  • Winding pairs and the fundamental group of the torus
  • Deformation theorem for torus loops
  • Integer matrices acting on the torus
  • Classifying five maps: quotient, homeomorphism, or neither
  • Comparing seven spaces with four invariants
  • Fundamental group of the sphere is trivial
  • Course synthesis and open problems
  • Topological definition of continuity
  • Inverse-image set algebra
  • Local neighborhood criterion for continuity
  • Equivalence with the epsilon-delta condition
  • Closed-set and basis criteria for continuity
  • Continuity of standard constructions
  • Closure inclusion under continuous maps
  • Homeomorphisms and topological equivalence

Syllabus

  1. Spaces, Open Sets and Continuity

    1. L1 · Distance, Neighborhoods and Open Sets
    2. L2 · Topological Spaces and Bases
    3. L3 · Closure, Interior, Boundary and Density
    4. L4 · Subspaces and Relative Topology
    5. L5 · Continuity and Homeomorphisms

  2. Building New Spaces

    1. L6 · Sequences and Countability
    2. L7 · Products and Coordinatewise Continuity
    3. L8 · Quotients and Gluing Spaces

  3. Connectedness

    1. L9 · Connectedness and the Intermediate Value Principle
    2. L10 · Paths, Components and Local Connectedness

  4. Compactness

    1. L11 · Compactness Through Open Covers
    2. L12 · Compactness in Euclidean Space
    3. L13 · Compact Spaces, Hausdorff Spaces and Closed Maps
    4. L14 · Compactness, Completeness and Total Boundedness
    5. L15 · Separation Axioms and Countability Revisited

  5. Homotopy and the Fundamental Group

    1. L16 · Homotopy and Deformation Retractions
    2. L17 · Loops and the Fundamental Group
    3. L18 · The Circle, Lifting and Winding Number
    4. L19 · Retractions and the Fixed Point Theorem
    5. L20 · A Topology Laboratory and Proof Capstone