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 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
- 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
- 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
- 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
- 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
- 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
- Gieseking manifold − A cusped hyperbolic 3-manifold of finite volume.
- Horosphere
- Picard horn
- Seifert–Weber space
Paradoxical spaces
- Lakes of Wada − Three disjoint connected open sets of or 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
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 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
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
- Bohr compactification
- Eells–Kuiper manifold
- Projectively extended real line
- Stone–Čech compactification
- Wallman compactification
Topologies of uniform convergence
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
- 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
- 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
- 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
Probability
Other topologies
- 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
Citations
References
External links
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