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 ${\displaystyle (X,\tau ),}$ the cocountable extension topology on ${\displaystyle X}$ is the topology having as a subbasis the union of τ and the family of all subsets of ${\displaystyle X}$ whose complements in ${\displaystyle X}$ 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 ${\displaystyle S^{1}.}$
• Sierpiński space, also called the connected two-point set − A 2-point set ${\displaystyle \{0,1\}}$ with the particular point topology ${\displaystyle \{\varnothing ,\{1\},\{0,1\}\}.}$

Integers

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

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 ${\displaystyle \mathbb {R} ^{2}}$ or ${\displaystyle (0,1)^{2}}$ 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.

• 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 ${\displaystyle [0,1]^{2}\setminus \mathbb {Q} ^{2},}$ where ${\displaystyle [0,1]^{2}\cap \mathbb {Q} ^{2}}$ 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 − ${\displaystyle [0,1/1]\times [0,1/2]\times [0,1/3]\times \cdots }$ 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 ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle x\in X}$ be a non-isolated point of ${\displaystyle X,}$ let ${\displaystyle d\not \in X}$ be arbitrary, and let ${\displaystyle Y=X\cup \{d\}.}$ Then ${\displaystyle \tau =\{V\subseteq Y:{\text{ either }}V{\text{ or }}(V\setminus \{d\})\cup \{x\}{\text{ is an open subset of }}X\}}$ is a topology on ${\displaystyle Y}$ and ${\displaystyle x}$ and ${\displaystyle d}$ have the same neighborhood filters in ${\displaystyle Y.}$ In this way, ${\displaystyle x}$ 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 ${\displaystyle X}$ that is homeomorphic to ${\displaystyle X\times X.}$
• 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 – Lie groups and their associated Lie algebras

Citations

1. Wilansky 2008, p. 35.