Topology

Explore spaces and continuity; global properties and constructions; separation and invariants. Concepts and connections are available independently of video coverage.

Prerequisites and connections

Sets
A set is a collection of distinct objects; membership is different from inclusion.
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Spaces and continuity

Topological spaces
A topology specifies open sets closed under arbitrary unions and finite intersections.
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Open and closed sets
Closed sets have open complements; a set can be both open and closed.
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Topological continuity
A map is continuous when inverse images of open sets are open.
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Homeomorphisms
A homeomorphism is a bijection continuous in both directions.
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Global properties and constructions

Connectedness
A connected space cannot be separated into two disjoint nonempty open pieces.
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Path connectedness
Every pair of points in a path-connected space can be joined by a continuous path.
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Compactness
A space is compact when every open cover admits a finite subcover.
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Product and quotient topologies
Product and quotient constructions create topologies compatible with coordinate maps or identifications.
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Separation and invariants

Hausdorff spaces
Distinct points admit disjoint neighborhoods in a Hausdorff space.
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Topological manifolds
A manifold locally resembles Euclidean space with suitable separation and countability conditions.
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Homotopy
A homotopy continuously deforms one map into another.
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Fundamental group
Based loops modulo endpoint-fixed homotopy form a group under concatenation.
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Curriculum scope reference