# 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 $(X,\tau ),$ the cocountable extension topology on $X$ is the topology having as a subbasis the union of τ and the family of all subsets of $X$ whose complements in $X$ 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 T$_0$. However, from the viewpoint of algebraic topology, it has the remarkable property that it is indistinguishable from the circle $S^{1}.$
* Sierpiński space, also called the connected two-point set − A 2-point set $\{0,1\}$ with the particular point topology $\{\varnothing ,\{1\},\{0,1\}\}.$

## Integers

* Arens–Fort space − A Hausdorff, regular, normal space that is not first-countable or compact. It has an element (i.e. $p:=(0,0)$) for which there is no sequence in $X\setminus \{p\}$ that converges to $p$ but there is a sequence $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ in $X\setminus \{(0,0)\}$ such that $(0,0)$ is a cluster point of $x_{\bullet }.$
* Arithmetic progression topologies
* The Baire space − $\mathbb {N} ^{\mathbb {N} }$ with the product topology, where $\mathbb {N}$ 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 $[0,1]$ 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 $[0,1]$ 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
* E$_8$ manifold − A topological manifold that does not admit a smooth structure.
* Euclidean topology − The natural topology on Euclidean space $\mathbb {R} ^{n}$ induced by the Euclidean metric, which is itself induced by the Euclidean norm.
** Real line − $\mathbb {R}$
** Unit interval − $[0,1]$
* 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 T$_1$ 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 $\mathbb {R} ^{3}.$

### 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 $\mathbb {R} ^{2}$ or $(0,1)^{2}$ 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.
* Alexandroff plank
* Appert topology − A Hausdorff, perfectly normal (T$_6$), zero-dimensional space that is countable, but neither first countable, locally compact, nor countably compact.
* Arens square
* Bullet-riddled square - The space $[0,1]^{2}\setminus \mathbb {Q} ^{2},$ where $[0,1]^{2}\cap \mathbb {Q} ^{2}$ 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 − $[0,1/1]\times [0,1/2]\times [0,1/3]\times \cdots$ 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 T$_0$ but not T$_1$.
* 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 $\mathbb {R}$ 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 $x\in X$ be a non-isolated point of $X,$ let $d\not \in X$ be arbitrary, and let $Y=X\cup \{d\}.$ Then $\tau =\{V\subseteq Y:{\text{ either }}V{\text{ or }}(V\setminus \{d\})\cup \{x\}{\text{ is an open subset of }}X\}$ is a topology on $Y$ and $x$ and $d$ have the same neighborhood filters in $Y.$ In this way, $x$ 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 $X$ that is homeomorphic to $X\times X.$
* Half-disk topology
* Hedgehog space
* Partition topology
* Zariski topology