Main page: List of topology topics

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.

Widely known topologies

• The Baire space − [math]\displaystyle{ \N^{\N} }[/math] with the product topology, where [math]\displaystyle{ \N }[/math] 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 [math]\displaystyle{ [0, 1] }[/math] with remarkable properties.
  • Cantor dust
• Discrete topology − All subsets are open.
• Euclidean topology − The natural topology on Euclidean space [math]\displaystyle{ \R^n }[/math] 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.
• Klein bottle
• Real projective line
• Torus
  • 3-torus
  • Solid torus

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. [math]\displaystyle{ p := (0, 0) }[/math]) for which there is no sequence in [math]\displaystyle{ X \setminus \{ p \} }[/math] that converges to [math]\displaystyle{ p }[/math] but there is a sequence [math]\displaystyle{ x_{\bull} = \left( x_i \right)_{i=1}^{\infty} }[/math] in [math]\displaystyle{ X \setminus \{ (0, 0) \} }[/math] such that [math]\displaystyle{ (0, 0) }[/math] is a cluster point of [math]\displaystyle{ x_{\bull}. }[/math]
• Branching line − A non-Hausdorff manifold.
• Bullet-riddled square - The space [math]\displaystyle{ [0, 1]^2 \setminus \Q^2, }[/math] where [math]\displaystyle{ [0, 1]^2 \cap \Q^2 }[/math] 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
• Lens space
• 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.
• Smith–Volterra–Cantor set − A closed nowhere dense (and thus meagre) subset of the unit interval [math]\displaystyle{ [0, 1] }[/math] that has positive Lebesgue measure.
• Sorgenfrey line, which is [math]\displaystyle{ \R }[/math] 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
• Whitehead manifold − An open 3-manifold that is contractible, but not homeomorphic to [math]\displaystyle{ \R^3. }[/math]

Hyperbolic geometry

• Gieseking manifold − A cusped hyperbolic 3-manifold of finite volume.
• Horosphere
  • Horocycle
• Picard horn
• Seifert–Weber space

Paradoxical spaces

• Gabriel's horn − It has infinite surface area but finite volume.

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.

Unique

• Hantzsche–Wendt manifold − A compact, orientable, flat 3-manifold. It is the only closed flat 3-manifold with first Betti number zero.

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
• Eells–Kuiper manifold
• Stone–Čech compactification
  • Stone topology
• 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

Functional analysis

• Auxiliary normed spaces
• Finest locally convex topology
• Finest vector topology
• Mackey topology
• Polar topology

Operator topologies

• Dual topology
• Norm topology
• Operator topologies
• Pointwise convergence
  • Weak convergence (Hilbert space)
  • Weak* 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

Other induced topologies

• Box topology
• Duplication of a point: Let [math]\displaystyle{ x \in X }[/math] be a non-isolated point of [math]\displaystyle{ X, }[/math] let [math]\displaystyle{ d \not\in X }[/math] be arbitrary, and let [math]\displaystyle{ Y = X \cup \{ d \}. }[/math] Then [math]\displaystyle{ \tau = \{ V \subseteq Y : \text{ either } V \text{ or } ( V \setminus \{ d \}) \cup \{ x \} \text{ is an open subset of } X \} }[/math] is a topology on [math]\displaystyle{ Y }[/math] and x and d have the same neighborhood filters in [math]\displaystyle{ Y. }[/math] In this way, x has been duplicated.^[1]

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 [math]\displaystyle{ (X, \tau), }[/math] 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
• 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 [math]\displaystyle{ X }[/math] that is homeomorphic to [math]\displaystyle{ X \times X. }[/math]
• 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 [math]\displaystyle{ S^1. }[/math]
• 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 hereditarily separable but not metrizable. Its metrizable subspaces are all countable.
• Zariski topology

See also

• Counterexamples in Topology – Book by Lynn Steen
• List of Banach spaces
• List of manifolds
• List of topology topics
• Lists of mathematics topics – None
• Natural topology – Notion in topology

Citations

References

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