• Source: List of topologies

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 they 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.

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




Citations






References






External links


π-Base: An Interactive Encyclopedia of Topological Spaces

Kata Kunci Pencarian:

• List of topologies
• Topology
• Operator topologies
• Order topology
• Euclidean topology
• List of topology topics
• Upper topology
• Cocountable topology
• Trivial topology
• List of geometric topology topics