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
[edit]- 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
[edit]- Cocountable topology
- Given a topological space the cocountable extension topology on is the topology having as a subbasis the union of τ and the family of all subsets of whose complements in are countable.
- Cofinite topology
- Double-pointed cofinite topology
- Ordinal number topology
- Pseudo-arc
- Ran space
- Tychonoff plank
Finite spaces
[edit]- 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
- Sierpiński space, also called the connected two-point set − A 2-point set with the particular point topology
Integers
[edit]- Arens–Fort space − A Hausdorff, regular, normal space that is not first-countable or compact. It has an element (i.e. ) for which there is no sequence in that converges to but there is a sequence in such that is a cluster point of
- Arithmetic progression topologies
- The Baire space − with the product topology, where denotes the natural numbers endowed with the discrete topology. It is the space of all sequences of natural numbers.
- Divisor topology
- Partition topology
Fractals and Cantor set
[edit]- Apollonian gasket
- Cantor set − A subset of the closed interval with remarkable properties.
- 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 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
[edit]- Alexandrov topology
- Lexicographic order topology on the unit square
- Order topology
- 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
[edit]- 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 induced by the Euclidean metric, which is itself induced by the Euclidean norm.
- 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
- Unknot
- Whitehead manifold − An open 3-manifold that is contractible, but not homeomorphic to
Hyperbolic geometry
[edit]- Gieseking manifold − A cusped hyperbolic 3-manifold of finite volume.
- Horosphere
- Picard horn
- Seifert–Weber space
Paradoxical spaces
[edit]- Lakes of Wada − Three disjoint connected open sets of or that they all have the same boundary.
Unique
[edit]- 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
[edit]Embeddings and maps between spaces
[edit]- 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)
[edit]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 where 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 − 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 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
[edit]Natural topologies
[edit]List of natural topologies.
- Adjunction space
- Disjoint union (topology)
- Extension topology
- Initial topology
- Final topology
- Product topology
- Quotient topology
- Subspace topology
- Weak topology
Compactifications
[edit]Compactifications include:
- Alexandroff extension
- Bohr compactification
- Eells–Kuiper manifold
- Projectively extended real line
- Stone–Čech compactification
- Wallman compactification
Topologies of uniform convergence
[edit]This lists named topologies of uniform convergence.
- Compact-open topology
- Interlocking interval topology
- Modes of convergence (annotated index)
- Operator topologies
- Pointwise convergence
- Polar topology
- Strong dual topology
- Topologies on spaces of linear maps
Other induced topologies
[edit]- Box topology
- Compact complement topology
- Duplication of a point: Let be a non-isolated point of let be arbitrary, and let Then is a topology on and and have the same neighborhood filters in In this way, has been duplicated.[1]
- Extension topology
Functional analysis
[edit]- Auxiliary normed spaces
- Finest locally convex topology
- Finest vector topology
- Helly space
- Mackey topology
- Polar topology
- Vague topology
Operator topologies
[edit]- Dual topology
- Norm topology
- Operator topologies
- Pointwise convergence
- 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
[edit]- Inductive tensor product
- Injective tensor product
- Projective tensor product
- Tensor product of Hilbert spaces
- Topological tensor product
Probability
[edit]Other topologies
[edit]- Erdős space − A Hausdorff, totally disconnected, one-dimensional topological space that is homeomorphic to
- Half-disk topology
- Hedgehog space
- Partition topology
- Zariski topology
See also
[edit]- 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
Citations
[edit]- ^ Wilansky 2008, p. 35.
References
[edit]- Adams, Colin; Franzosa, Robert (2009). Introduction to Topology: Pure and Applied. New Delhi: Pearson Education. ISBN 978-81-317-2692-1. OCLC 789880519.
- Arkhangel'skii, Alexander Vladimirovich; Ponomarev, V.I. (1984). Fundamentals of General Topology: Problems and Exercises. Mathematics and Its Applications. Vol. 13. Dordrecht Boston: D. Reidel. ISBN 978-90-277-1355-1. OCLC 9944489.
- Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
- Bourbaki, Nicolas (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. Éléments de mathématique. Vol. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063.
- Comfort, William Wistar; Negrepontis, Stylianos (1974). The Theory of Ultrafilters. Vol. 211. Berlin Heidelberg New York: Springer-Verlag. ISBN 978-0-387-06604-2. OCLC 1205452.
- Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303.
- Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011.
- Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
- Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
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- Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
- Joshi, K. D. (1983). Introduction to General Topology. New York: John Wiley and Sons Ltd. ISBN 978-0-85226-444-7. OCLC 9218750.
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- Schubert, Horst (1968). Topology. London: Macdonald & Co. ISBN 978-0-356-02077-8. OCLC 463753.
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