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Topology

∪∩

Topology

Point-set topology and introduction to algebraic topology

196Modules

9Units

Being rebuilt

This course is still in our original format. All the content is here and free to use, but it has not yet been through the rebuild that gives our newer courses their interactive animations and cleaner visuals — so you may hit some rough edges. Spot an error? Send us a note and we will fix it.

Recommended preparation

Real Analysis is the natural lead-in — topology generalizes the open sets, limits, and continuity you meet there.

Rubber Sheet GeometryINTRO

The Coffee Cup and the DonutINTRO

What is Topology?INTRO

Continuous DeformationINTRO

Review of Set TheoryLESSON

Open Sets - Building IntuitionLESSON

Open Set ExplorerACTIVITY

Definition of a Topological SpaceLESSON

Verifying Topology AxiomsEXAMPLE

The Standard Topology on Real NumbersEXAMPLE

Discrete and Indiscrete TopologiesEXAMPLE

Topology BuilderACTIVITY

Topology Axioms PracticePRACTICE

Closed Sets and Their PropertiesLESSON

Open and Closed SetsPRACTICE

Neighborhoods and Interior PointsLESSON

Closure, Interior, and BoundaryLESSON

Finding Closure and BoundaryEXAMPLE

Closure and Boundary CalculatorACTIVITY

Closure and Interior ProblemsPRACTICE

Foundations of Topology SummarySUMMARY

Foundations Reference SheetREFERENCE

Foundations of Topology QuizQUIZ

Building Blocks of TopologyINTRO

Generating Topologies from Simple SetsINTRO

Definition of a BasisLESSON

Basis GeneratorACTIVITY

The Basis CriterionLESSON

Standard Basis for the Real LineEXAMPLE

Basis ProblemsPRACTICE

Generating a Topology from a BasisLESSON

Comparing Topologies - Finer and CoarserLESSON

Topology ComparatorACTIVITY

Comparing TopologiesPRACTICE

Subbasis and Generated TopologyLESSON

The Order TopologyLESSON

The Lower Limit TopologyEXAMPLE

The Product TopologyLESSON

Product Topology on IntervalsEXAMPLE

Product Topology VisualizerACTIVITY

The Subspace TopologyLESSON

Subspaces of the Real LineEXAMPLE

Product and Subspace TopologiesPRACTICE

Bases and Subbases SummarySUMMARY

Bases and Subbases ReferenceREFERENCE

Bases and Subbases QuizQUIZ

Stretching Without BreakingINTRO

When Shapes Are the SameINTRO

Continuous Functions in TopologyLESSON

Continuity at a PointLESSON

Proving ContinuityEXAMPLE

Continuity ProofsPRACTICE

Properties of Continuous MapsLESSON

Map Projections and ContinuityREAL WORLD

Continuous Data TransformationsREAL WORLD

Homeomorphisms - Topological IsomorphismsLESSON

Homeomorphisms Between IntervalsEXAMPLE

Constructing HomeomorphismsLESSON

Homeomorphism BuilderACTIVITY

Shape TransformerACTIVITY

Homeomorphism ProblemsPRACTICE

Discovering Topological InvariantsINTRO

Topological InvariantsLESSON

Circle vs Line SegmentEXAMPLE

Quotient Spaces and Quotient MapsLESSON

Constructing the Circle as a QuotientEXAMPLE

Quotient Space ConstructorACTIVITY

Quotient Space ProblemsPRACTICE

Continuity and Homeomorphisms SummarySUMMARY

Continuity Reference SheetREFERENCE

Continuity and Homeomorphisms QuizQUIZ

One-Piece SpacesINTRO

Definition of Connected SpacesLESSON

The Real Line is ConnectedEXAMPLE

The Rationals are DisconnectedEXAMPLE

Connectedness TesterACTIVITY

Connected SubsetsLESSON

Connectedness Under Continuous MapsLESSON

Connectedness ProofsPRACTICE

Can You Walk There?INTRO

Path ConnectednessLESSON

Path FinderACTIVITY

Path Connectedness ProblemsPRACTICE

Path Connected vs ConnectedLESSON

The Topologist's Sine CurveEXAMPLE

Finding Pieces of a SpaceINTRO

Connected ComponentsLESSON

Finding Connected ComponentsEXAMPLE

Component FinderACTIVITY

Component ProblemsPRACTICE

Local ConnectednessLESSON

Network ConnectivityREAL WORLD

Connectedness SummarySUMMARY

Connectedness Reference SheetREFERENCE

Connectedness QuizQUIZ

Finite from InfiniteINTRO

Why Compactness MattersINTRO

Covers and SubcoversINTRO

Open Covers and SubcoversLESSON

Cover ExplorerACTIVITY

Open Cover ProblemsPRACTICE

Definition of CompactnessLESSON

Compact Subsets of the Real LineLESSON

Closed Intervals are CompactEXAMPLE

Open Intervals are Not CompactEXAMPLE

The Cantor Set is CompactEXAMPLE

The Heine-Borel TheoremLESSON

Heine-Borel VisualizerACTIVITY

Heine-Borel ApplicationsPRACTICE

Properties of Compact SpacesLESSON

Compactness and Continuous FunctionsLESSON

Continuous Images of Compact SetsEXAMPLE

Compactness TesterACTIVITY

Compactness ProofsPRACTICE

The Extreme Value TheoremLESSON

Compactness in OptimizationREAL WORLD

Modeling Finite ResourcesREAL WORLD

Sequential CompactnessLESSON

Limit Point CompactnessLESSON

Local CompactnessLESSON

Compactness SummarySUMMARY

Compactness Reference SheetREFERENCE

Compactness QuizQUIZ

Distinguishing PointsINTRO

How Separated is Separated?INTRO

T0 (Kolmogorov) SpacesLESSON

T1 SpacesLESSON

Hausdorff (T2) SpacesLESSON

Hausdorff Space ExamplesEXAMPLE

Non-Hausdorff SpacesEXAMPLE

Hausdorff ProofsPRACTICE

Regular SpacesLESSON

Normal SpacesLESSON

Proving a Space is NormalEXAMPLE

Normal Space ProblemsPRACTICE

Urysohn's LemmaLESSON

Applying Urysohn's LemmaEXAMPLE

Urysohn Function BuilderACTIVITY

Tietze Extension TheoremLESSON

The Separation HierarchyLESSON

Separation ClassifierACTIVITY

Separation Axiom ProblemsPRACTICE

Separation Axioms SummarySUMMARY

Separation Axioms ReferenceREFERENCE

Separation Axioms QuizQUIZ

Measuring DistanceINTRO

Different Ways to MeasureINTRO

Definition of a Metric SpaceLESSON

The Euclidean MetricEXAMPLE

The Taxicab MetricEXAMPLE

The Discrete MetricEXAMPLE

Metric Space ProblemsPRACTICE

GPS and Distance MetricsREAL WORLD

The Metric TopologyLESSON

Open Ball VisualizerACTIVITY

Metric ExplorerACTIVITY

Equivalent MetricsLESSON

Metrizable SpacesLESSON

Complete Metric SpacesLESSON

Proving CompletenessEXAMPLE

Completeness TesterACTIVITY

Completeness ProblemsPRACTICE

The Baire Category TheoremLESSON

Countability AxiomsLESSON

Metrizability ProblemsPRACTICE

Metric Spaces SummarySUMMARY

Metric Spaces ReferenceREFERENCE

Metric Spaces QuizQUIZ

Loops and HolesINTRO

Counting Holes in SpacesINTRO

Rubber Band Around a HoleINTRO

Homotopy of PathsLESSON

Loop DrawerACTIVITY

Homotopy ProblemsPRACTICE

The Fundamental GroupLESSON

Fundamental Group of the PlaneEXAMPLE

Fundamental Group CalculatorACTIVITY

Simply Connected SpacesLESSON

The Fundamental Group of the CircleLESSON

Fundamental Group of the TorusEXAMPLE

Fundamental Group ProblemsPRACTICE

Introduction to Covering SpacesLESSON

Van Kampen's TheoremLESSON

DNA and TopologyREAL WORLD

Robotics and Path PlanningREAL WORLD

The Euler CharacteristicLESSON

Euler Characteristic of PolyhedraEXAMPLE

Euler Characteristic CalculatorACTIVITY

Euler Characteristic ProblemsPRACTICE

Introduction to Algebraic Topology SummarySUMMARY

Algebraic Topology ReferenceREFERENCE

Algebraic Topology QuizQUIZ

Comprehensive Topology Practice TestQUIZ

Topology Problem Solving Strategy GuideSUMMARY

Topology Master Reference CompendiumREFERENCE

Topology GlossaryFORMULA SHEET