From Wikipedia, the free encyclopedia
List of concrete topologies and topological spaces
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
Cardinality and ordinals
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
Fractals and Cantor set
Orders
Manifolds and complexes
Hyperbolic geometry
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
Related or similar to manifolds
Embeddings and maps between spaces
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 .
Compactifications
Compactifications include:
Topologies of uniform convergence
This lists named topologies of
uniform convergence .
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
Operator topologies
Tensor products
Probability
Other topologies
See also
Citations
References
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External links