Topology

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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

• Basis of topology
• Subbasis of topology
• Subspace topology
• Product topology
• Quotient topology

Metric Spaces

Basic Definitions

• Metric
• Metric space
• Metric-induced topology
• Equivalent metrics
• Open ball
• Closed ball
• Metric sphere
• Diameter
• Bounded set

Completeness

• Cauchy sequence
• Complete metric space
• Convergent sequence
• Convergent ⟹ Cauchy
• Cauchy ⟹ bounded

Total Boundedness

• Totally bounded set
• ε-net
• Totally bounded ⟹ Cauchy subsequence

Continuity

• Continuous map
• Homeomorphism
• Uniformly continuous map
• Lipschitz continuity
• Hölder continuity
• Isometry

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

Connectedness

Definitions

• Separated sets
• Connected set
• Connected component
• Path
• Path-connected set
• Curve

Theorems

• Continuous image of connected is connected
• Connected subsets of ℝ are intervals
• Connectedness criteria in ℝ
• Image of compact connected is interval

Separation Axioms

• T₀ space
• T₁ space
• Hausdorff space (T₂)
• Uniqueness of limits in Hausdorff
• Sequential characterization of closed
• Sequential characterization of closure

Baire Category

Definitions

• Nowhere dense set
• Meager set (first category)
• Residual set
• Baire space

Theorems

• Baire category theorem
• Complete metric space is Baire
• Intersection of dense open is dense
• Category argument template

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