kb/data/en.wikipedia.org/wiki/List_of_topologies-0.md

title	chunk	source	category	tags	date_saved	instance
List of topologies	1/2	https://en.wikipedia.org/wiki/List_of_topologies	reference	science, encyclopedia	2026-05-05T08:17:38.117299+00:00	kb-cron

The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.

== Discrete and indiscrete == Discrete topology − All subsets are open. Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open.

== Cardinality and ordinals ==

Cocountable topology Given a topological space

    (
    X
    ,
    τ
    )
    ,
  

{\displaystyle (X,\tau ),}

the cocountable extension topology on

    X
  

{\displaystyle X}

is the topology having as a subbasis the union of τ and the family of all subsets of

    X
  

{\displaystyle X}

whose complements in

    X
  

{\displaystyle X}

are countable. Cofinite topology Double-pointed cofinite topology Ordinal number topology Pseudo-arc Ran space Tychonoff plank

=== Finite spaces === Discrete two-point space − The simplest example of a totally disconnected discrete space. Finite topological space Pseudocircle − A finite topological space on 4 elements that fails to satisfy any separation axiom besides T0. However, from the viewpoint of algebraic topology, it has the remarkable property that it is indistinguishable from the circle

      S
      
        1
      
    
    .
  

{\displaystyle S^{1}.}

Sierpiński space, also called the connected two-point set − A 2-point set

    {
    0
    ,
    1
    }
  

{\displaystyle \{0,1\}}

with the particular point topology

    {
    ∅
    ,
    {
    1
    }
    ,
    {
    0
    ,
    1
    }
    }
    .
  

{\displaystyle \{\varnothing ,\{1\},\{0,1\}\}.}

== Integers == Arens–Fort space − A Hausdorff, regular, normal space that is not first-countable or compact. It has an element (i.e.

    p
    :=
    (
    0
    ,
    0
    )
  

{\displaystyle p:=(0,0)}

) for which there is no sequence in

    X
    ∖
    {
    p
    }
  

{\displaystyle X\setminus \{p\}}

that converges to

    p
  

{\displaystyle p}

but there is a sequence

      x
      
        ∙
      
    
    =
    
      
        (
        
          x
          
            i
          
        
        )
      
      
        i
        =
        1
      
      
        ∞
      
    
  

{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}

in

    X
    ∖
    {
    (
    0
    ,
    0
    )
    }
  

{\displaystyle X\setminus \{(0,0)\}}

such that

    (
    0
    ,
    0
    )
  

{\displaystyle (0,0)}

is a cluster point of

      x
      
        ∙
      
    
    .
  

{\displaystyle x_{\bullet }.}

Arithmetic progression topologies The Baire space −

        N
      
      
        
          N
        
      
    
  

{\displaystyle \mathbb {N} ^{\mathbb {N} }}

with the product topology, where

      N
    
  

{\displaystyle \mathbb {N} }

denotes the natural numbers endowed with the discrete topology. It is the space of all sequences of natural numbers. Divisor topology Partition topology Deleted integer topology Odd–even topology

== Fractals and Cantor set ==

Apollonian gasket Cantor set − A subset of the closed interval

    [
    0
    ,
    1
    ]
  

{\displaystyle [0,1]}

with remarkable properties. Cantor dust Cantor space Koch snowflake Menger sponge Mosely snowflake Sierpiński carpet Sierpiński triangle Smith–Volterra–Cantor set, also called the fat Cantor set − A closed nowhere dense (and thus meagre) subset of the unit interval

    [
    0
    ,
    1
    ]
  

{\displaystyle [0,1]}

that has positive Lebesgue measure and is not a Jordan measurable set. The complement of the fat Cantor set in Jordan measure is a bounded open set that is not Jordan measurable.

== Orders ==

Alexandrov topology Lexicographic order topology on the unit square Order topology Lawson topology Poset topology Upper topology Scott topology Scott continuity Priestley space Roy's lattice space Split interval, also called the Alexandrov double arrow space and the two arrows space − All compact separable ordered spaces are order-isomorphic to a subset of the split interval. It is compact Hausdorff, hereditarily Lindelöf, and hereditarily separable but not metrizable. Its metrizable subspaces are all countable. Specialization (pre)order

== Manifolds and complexes ==

Branching line − A non-Hausdorff manifold. Double origin topology E8 manifold − A topological manifold that does not admit a smooth structure. Euclidean topology − The natural topology on Euclidean space

        R
      
      
        n
      
    
  

{\displaystyle \mathbb {R} ^{n}}

induced by the Euclidean metric, which is itself induced by the Euclidean norm. Real line −

      R
    
  

{\displaystyle \mathbb {R} }

Unit interval −

    [
    0
    ,
    1
    ]
  

{\displaystyle [0,1]}

Extended real number line Fake 4-ball − A compact contractible topological 4-manifold. House with two rooms − A contractible, 2-dimensional simplicial complex that is not collapsible. Klein bottle Lens space Line with two origins, also called the bug-eyed line − It is a non-Hausdorff manifold. It is locally homeomorphic to Euclidean space and thus locally metrizable (but not metrizable) and locally Hausdorff (but not Hausdorff). It is also a T1 locally regular space but not a semiregular space. Prüfer manifold − A Hausdorff 2-dimensional real analytic manifold that is not paracompact. Real projective line Torus 3-torus Solid torus Unknot Whitehead manifold − An open 3-manifold that is contractible, but not homeomorphic to

        R
      
      
        3
      
    
    .
  

{\displaystyle \mathbb {R} ^{3}.}

=== Hyperbolic geometry === Gieseking manifold − A cusped hyperbolic 3-manifold of finite volume. Horosphere Horocycle Picard horn Seifert–Weber space

=== Paradoxical spaces === Lakes of Wada − Three disjoint connected open sets of

        R
      
      
        2
      
    
  

{\displaystyle \mathbb {R} ^{2}}

or

    (
    0
    ,
    1
    
      )
      
        2
      
    
  

{\displaystyle (0,1)^{2}}

that all have the same boundary.

=== Unique === Hantzsche–Wendt manifold − A compact, orientable, flat 3-manifold. It is the only closed flat 3-manifold with first Betti number zero.

=== Related or similar to manifolds === Dogbone space Dunce hat (topology) Hawaiian earring Long line (topology) Rose (topology)

== Embeddings and maps between spaces == Alexander horned sphere − A particular embedding of a sphere into 3-dimensional Euclidean space. Antoine's necklace − A topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. Irrational winding of a torus/Irrational cable on a torus Knot (mathematics) Linear flow on the torus Space-filling curve Torus knot Wild knot

== Counter-examples (general topology) == The following topologies are a known source of counterexamples for point-set topology.