List of general topology topics

Basic concepts

• Topological space
• Topological property
• Open set, closed set
• *Clopen set
• *Closure
• *Boundary
• *Dense
• *G-delta set, F-sigma set
• *closeness
• *neighbourhood
• Continuity
• *Homeomorphism
• *Local homeomorphism
• *Open and closed maps
• *Germ
• Base, subbase
• Open cover
• Covering space
• Atlas

  Limits

• Limit point
• Net
• Filter
• Ultrafilter

  [Topological properties]

• Baire category theorem
• *Nowhere dense
• *Baire space
• *Banach–Mazur game
• *Meagre set
• *Comeagre set

  Compactness and countability

• Compact space
• *Relatively compact subspace
• *Heine–Borel theorem
• *Tychonoff's theorem
• *Finite intersection property
• *Compactification
• *Measure of non-compactness
• Paracompact space
• Locally compact space
• Compactly generated space
• Axiom of countability
• Sequential space
• First-countable space
• Second-countable space
• Separable space
• Lindelöf space
• Sigma-compact space

  Connectedness

• Connected space

  [Separation axioms]

• T0 space
• T1 space
• Hausdorff space
• *Completely Hausdorff space
• Regular space
• Tychonoff space
• Normal space
• Urysohn's lemma
• Tietze extension theorem
• Paracompact
• Separated sets

  Topological constructions

• direct sum and the dual construction product
• subspace and the dual construction quotient
• Topological tensor product

  Examples

• Discrete space
• *Locally constant function
• Trivial topology
• Cofinite topology
• Finer topology
• Product topology
• *Restricted product
• Quotient space
• Unit interval
• Continuum
• Extended real number line
• Long line
• Sierpinski space
• Cantor set, Cantor space, Cantor cube
• Space-filling curve
• Topologist's sine curve
• Uniform norm
• Weak topology
• Strong topology
• Hilbert cube
• Lower limit topology
• Sorgenfrey plane
• Real tree
• Compact-open topology
• Zariski topology
• Kuratowski closure axioms
• Unicoherent
• Solenoid

  [Uniform space]s

• Uniform continuity
• Lipschitz continuity
• Uniform isomorphism
• Uniform property
• Uniformly connected space

  [Metric space]s

• Metric topology
• Manhattan distance
• Ultrametric space
• *P-adic numbers, p-adic analysis
• Open ball
• Bounded subset
• Pointwise convergence
• Metrization theorems
• Complete space
• *Cauchy sequence
• *Banach fixed point theorem
• Polish space
• Hausdorff distance
• Intrinsic metric
• Category of metric spaces

  Topology and [order theory]

• Stone duality
• *Stone's representation theorem for Boolean algebras
• Specialization order
• Sober space
• Spectral space
• Alexandrov topology
• Upper topology
• Scott topology
• *Scott continuity
• Lawson topology

  [Descriptive set theory]

[Dimension theory]

• Inductive dimension
• Lebesgue covering dimension
• Lebesgue's number lemma

  [Combinatorial topology]

• Polytope
• Simplex
• Simplicial complex
• CW complex
• Manifold
• Triangulation
• Barycentric subdivision
• Sperner's lemma
• Simplicial approximation theorem
• Nerve of an open covering

  Foundations of [algebraic topology]

• Simply connected
• Semi-locally simply connected
• Path
• Homotopy
• Homotopy lifting property
• Pointed space
• Wedge sum
• smash product
• Cone
• Adjunction space

  Topology and algebra

• Topological algebra
• Topological group
• Topological ring
• Topological vector space
• Topological module
• Topological abelian group
• Properly discontinuous
• Sheaf space