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 T0. 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 $$x_\bull = \left(x_i\right)_{i=1}^\infty$$ in $$X \setminus \{(0, 0)\}$$ such that $$(0, 0)$$ is a cluster point of $$x_\bull.$$
 * Arithmetic progression topologies
 * The Baire space − $$\N^{\N}$$ with the product topology, where $$\N$$ 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 − 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 ' and the ' − 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 $$\Reals^n$$ induced by the Euclidean metric, which is itself induced by the Euclidean norm.
 * Real line − $$\Reals$$
 * 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  − 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 $$\Reals^3.$$

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 $$\Reals^2$$ or $$(0, 1)^2$$ 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 $$[0, 1]^2 \setminus \Q^2,$$ where $$[0, 1]^2 \cap \Q^2$$ 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  − 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 $$\Reals$$ 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

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