# List of general topology topics

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