kb/data/en.wikipedia.org/wiki/List_of_topologies-1.md

title	chunk	source	category	tags	date_saved	instance
List of topologies	2/2	https://en.wikipedia.org/wiki/List_of_topologies	reference	science, encyclopedia	2026-05-05T08:17:38.117299+00:00	kb-cron

Alexandroff plank Appert topology − A Hausdorff, perfectly normal (T6), zero-dimensional space that is countable, but neither first countable, locally compact, nor countably compact. Arens square Bullet-riddled square - The space

    [
    0
    ,
    1
    
      ]
      
        2
      
    
    ∖
    
      
        Q
      
      
        2
      
    
    ,
  

{\displaystyle [0,1]^{2}\setminus \mathbb {Q} ^{2},}

where

    [
    0
    ,
    1
    
      ]
      
        2
      
    
    ∩
    
      
        Q
      
      
        2
      
    
  

{\displaystyle [0,1]^{2}\cap \mathbb {Q} ^{2}}

is the set of bullets. Neither of these sets is Jordan measurable although both are Lebesgue measurable. Cantor tree Comb space Dieudonné plank Double origin topology Dunce hat (topology) Either–or topology Excluded point topology − A topological space where the open sets are defined in terms of the exclusion of a particular point. Fort space Half-disk topology Hilbert cube −

    [
    0
    ,
    1
    
      /
    
    1
    ]
    ×
    [
    0
    ,
    1
    
      /
    
    2
    ]
    ×
    [
    0
    ,
    1
    
      /
    
    3
    ]
    ×
    ⋯
  

{\displaystyle [0,1/1]\times [0,1/2]\times [0,1/3]\times \cdots }

with the product topology. Infinite broom Integer broom topology K-topology Knaster–Kuratowski fan Long line (topology) Moore plane, also called the Niemytzki plane − A first countable, separable, completely regular, Hausdorff, Moore space that is not normal, Lindelöf, metrizable, second countable, nor locally compact. It also an uncountable closed subspace with the discrete topology. Nested interval topology Overlapping interval topology − Second countable space that is T0 but not T1. Particular point topology − Assuming the set is infinite, then contains a non-closed compact subset whose closure is not compact and moreover, it is neither metacompact nor paracompact. Rational sequence topology Sorgenfrey line, which is

      R
    
  

{\displaystyle \mathbb {R} }

endowed with lower limit topology − It is Hausdorff, perfectly normal, first-countable, separable, paracompact, Lindelöf, Baire, and a Moore space but not metrizable, second-countable, σ-compact, nor locally compact. Sorgenfrey plane, which is the product of two copies of the Sorgenfrey line − A Moore space that is neither normal, paracompact, nor second countable. Topologist's sine curve Tychonoff plank Vague topology Warsaw circle

== Topologies defined in terms of other topologies ==

=== Natural topologies === List of natural topologies.

Adjunction space Disjoint union (topology) Extension topology Initial topology Final topology Product topology Quotient topology Subspace topology Weak topology

=== Compactifications === Compactifications include:

Alexandroff extension Projectively extended real line Bohr compactification Eells–Kuiper manifold Projectively extended real line Stone–Čech compactification Stone topology Stone–Čech remainder Wallman compactification

=== Topologies of uniform convergence === This lists named topologies of uniform convergence.

Compact-open topology Loop space Interlocking interval topology Modes of convergence (annotated index) Operator topologies Pointwise convergence Weak convergence (Hilbert space) Weak* topology Polar topology Strong dual topology Topologies on spaces of linear maps

=== Other induced topologies === Box topology Compact complement topology Duplication of a point: Let

    x
    ∈
    X
  

{\displaystyle x\in X}

be a non-isolated point of

    X
    ,
  

{\displaystyle X,}

let

    d
    ∉
    X
  

{\displaystyle d\not \in X}

be arbitrary, and let

    Y
    =
    X
    ∪
    {
    d
    }
    .
  

{\displaystyle Y=X\cup \{d\}.}

Then

    τ
    =
    {
    V
    ⊆
    Y
    :
    
       either 
    
    V
    
       or 
    
    (
    V
    ∖
    {
    d
    }
    )
    ∪
    {
    x
    }
    
       is an open subset of 
    
    X
    }
  

{\displaystyle \tau =\{V\subseteq Y:{\text{ either }}V{\text{ or }}(V\setminus \{d\})\cup \{x\}{\text{ is an open subset of }}X\}}

is a topology on

    Y
  

{\displaystyle Y}

and

    x
  

{\displaystyle x}

and

    d
  

{\displaystyle d}

have the same neighborhood filters in

    Y
    .
  

{\displaystyle Y.}

In this way,

    x
  

{\displaystyle x}

has been duplicated. Extension topology

== Functional analysis == Auxiliary normed spaces Finest locally convex topology Finest vector topology Helly space Mackey topology Polar topology Vague topology

=== Operator topologies === Dual topology Norm topology Operator topologies Pointwise convergence Weak convergence (Hilbert space) Weak* topology Polar topology Strong dual space Strong operator topology Topologies on spaces of linear maps Ultrastrong topology Ultraweak topology/weak-* operator topology Weak operator topology

=== Tensor products === Inductive tensor product Injective tensor product Projective tensor product Tensor product of Hilbert spaces Topological tensor product

== Probability == Émery topology

== Other topologies == Erdős space − A Hausdorff, totally disconnected, one-dimensional topological space

    X
  

{\displaystyle X}

that is homeomorphic to

    X
    ×
    X
    .
  

{\displaystyle X\times X.}

Half-disk topology Hedgehog space Partition topology Zariski topology

== See also == Counterexamples in Topology – Book by Lynn Steen List of Banach spaces List of fractals by Hausdorff dimension List of manifolds List of topologies on the category of schemes List of topology topics Lists of mathematics topics Natural topology – Notion in topology Table of Lie groups – Lie groups and their associated Lie algebras

== Citations ==

== References ==

== External links == π-Base: An Interactive Encyclopedia of Topological Spaces