Section
Topology
Topological spaces, metric spaces, compactness, connectedness, and Baire category
Core idea
Click any term to expand its definition inline.
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Topological Spaces
Basic Definitions
- Topological space
- Topology
- Open set
- Closed set
- Neighborhood
- Interior
- Closure
- Limit point
- Derived set
- Dense set
Generating Topologies
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Metric Spaces
Basic Definitions
- Metric
- Metric space
- Metric-induced topology
- Equivalent metrics
- Open ball
- Closed ball
- Metric sphere
- Diameter
- Bounded set
Completeness
Total Boundedness
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Continuity
- Continuous map
- Homeomorphism
- Uniformly continuous map
- Lipschitz continuity
- Hölder continuity
- Isometry
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Compactness
Definitions
- Cover
- Open cover
- Refinement
- Compact set
- Sequentially compact set
- Relatively compact set
- Finite intersection property
Theorems
- Sequential compactness = compactness
- FIP characterization
- Compact ⟺ complete + totally bounded
- Heine–Borel theorem
- Bolzano–Weierstrass theorem
- Lebesgue number lemma
- Cantor intersection theorem
Properties
- Compact ⟹ bounded
- Compact ⟹ closed
- Compact ⟹ complete
- Compact ⟹ totally bounded
- Closed subset of compact is compact
- Compact in Hausdorff is closed
- Continuous image of compact is compact
- Continuous attains max/min on compact
- Bijection from compact to Hausdorff
- Compactness of graphs lemma
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Connectedness
Definitions
Theorems
- Continuous image of connected is connected
- Connected subsets of ℝ are intervals
- Connectedness criteria in ℝ
- Image of compact connected is interval
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Separation Axioms
- T₀ space
- T₁ space
- Hausdorff space (T₂)
- Uniqueness of limits in Hausdorff
- Sequential characterization of closed
- Sequential characterization of closure
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Baire Category
Definitions
Theorems
- Baire category theorem
- Complete metric space is Baire
- Intersection of dense open is dense
- Category argument template
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Uncategorized
- Boundary
- Closed sets are complements of open sets
- Continuity via open sets
- Continuous bijection from compact is a homeomorphism criterion
- Continuous function on a compact set is bounded
- Continuous functions on compact sets are bounded
- Convergence in product metric spaces
- Continuous function on a compact set is uniformly continuous
- Nested interval theorem
- Open sets form a topology
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Manifolds, groups, and homotopy
- Topological manifold
- Closed manifold
- Locally compact space
- Locally compact group
- Continuous group action
- Separable space
- Homotopy equivalence
- Homotopy sphere