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 have; for that, see List of general topology topics and Topological property.

Widely known 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.
• Cantor set − A subset of the closed interval ${\displaystyle [0,1]}$ with remarkable properties.
  • Cantor dust
• Discrete topology − All subsets are open.
• Euclidean topology − The natural topology topology on Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ induced by the Euclidean metric, which is itself induced by the Euclidean norm.
• Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open.

Counter-example topologies[]

The following topologies are a known source of counterexamples for point-set topology.

• Appert topology − A Hausdorff, perfectly normal (T₆), zero-dimensional space that is countable, but neither first countable, locally compact, nor countably compact.
• 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 ${\displaystyle x_{\bullet }.}$
• Branching line − A non-Hausdorff manifold.
• - 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.
• Comb space
• Dogbone space
• Dunce hat (topology)
• E₈ manifold − A topological manifold that does not admit a smooth structure.
• Excluded point topology − A topological space where the open sets are defined in terms of the exclusion of a particular point.
• Fort space
• House with two rooms − A contractible, 2-dimensional Simplicial complex that is not collapsible.
• Infinite broom
• Integer broom topology
• K-topology
• Lexicographic order topology on the unit square
• Line with two origins, also called the bug-eyed line − It is a non-Hausdorff manifold and a locally regular space but not a semiregular space.
• 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.
• 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.
• Prüfer manifold − A Hausdorff 2-dimensional real analytic manifold that is not paracompact.
• 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
• Warsaw circle
• Whitehead manifold − An open 3-manifold that is contractible, but not homeomorphic to ${\displaystyle \mathbb {R} ^{3}.}$

Pathological embeddings of 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.

Topologies defined in terms of other topologies[]

Natural topologies[]

List of natural topologies.

• Corona set
• Disjoint union (topology)
• Extension topology
• Initial topology
• Final topology
• Product topology
• Quotient topology
• Subspace topology
• Weak topology

Compactifications[]

• Alexandroff extension
  • Projectively extended real line
• Bohr compactification
• Stone–Čech compactification
  • Stone topology
• Wallman compactification

Other induced topologies[]

• Box 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 x and d have the same neighborhood filters in ${\displaystyle Y.}$ In this way, x has been duplicated.^[1]

Topologies of uniform convergence[]

This lists named topologies of uniform convergence.

• Compact-open topology
  • Loop space
• Interlocking interval topology
• Pointwise convergence
  • Weak convergence (Hilbert space)
  • Weak* topology
• Polar topology
• Strong dual topology

Fractal spaces[]

• Apollonian gasket
• Cantor set
• Koch snowflake
• Menger sponge
• Mosely snowflake
• Sierpiński carpet
• Sierpiński triangle

Topologies related to other structures[]

• Order topology

Other topologies[]

• Cantor space
• Cocountable topology
  • Given a topological space ${\displaystyle (X,\tau ),}$ the 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
• Discrete two-point space − The simplest example of a totally disconnected discrete space.
• Double-pointed cofinite topology
• Erdős space − A Hausdorff, totally disconnected, one-dimensional topological space ${\displaystyle X}$ that is homeomorphic to ${\displaystyle X\times X.}$
• Fake 4-ball − A compact contractible topological 4-manifold.
• Half-disk topology
• Hawaiian earring
• Hedgehog space
• Long line (topology)
• 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 ${\displaystyle S^{1}.}$
• Rose (topology)
• 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 but not metrizable. Its metrizable subspaces are all countable.
• Zariski topology

See also[]

• List of Banach spaces
• List of manifolds – Wikipedia list article
• List of topology topics – Wikipedia list article
• Natural topology – notion in topology

Citations[]

References[]