This is a list of general topology topics.

Basic concepts

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

Limits

• Limit point
• Net (topology)
• Filter (topology)
• 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 (topology)
• Extended real number line
• Long line (topology)
• 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 (mathematics)

Uniform spaces

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

Metric spaces

• 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 (pre)order
• Sober space
• Spectral space
• Alexandrov topology
• Upper topology
• Scott topology
  • Scott continuity
• Lawson topology

Descriptive set theory

• Polish Space
• Cantor space

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 (topology)
• Homotopy
• Homotopy lifting property
• Pointed space
• Wedge sum
• Smash product
• Cone (topology)
• Adjunction space

Topology and algebra

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