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 ( X , τ ) , {\displaystyle (X,\tau ),} the cocountable extension topology on X {\displaystyle X} is the topology having as a subbasis the union of τ and the family of all subsets of X {\displaystyle X} whose complements in X {\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 S 1 . {\displaystyle S^{1}.}
Sierpiński space, also called the connected two-point set − A 2-point set { 0 , 1 } {\displaystyle \{0,1\}} with the particular point topology { ∅ , { 1 } , { 0 , 1 } } . {\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. p := ( 0 , 0 ) {\displaystyle p:=(0,0)} ) for which there is no sequence in X ∖ { p } {\displaystyle X\setminus \{p\}} that converges to p {\displaystyle p} but there is a sequence x ∙ = ( x i ) i = 1 ∞ {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }} in X ∖ { ( 0 , 0 ) } {\displaystyle X\setminus \{(0,0)\}} such that ( 0 , 0 ) {\displaystyle (0,0)} is a cluster point of x ∙ . {\displaystyle x_{\bullet }.}
Arithmetic progression topologies The Baire space − N N {\displaystyle \mathbb {N} ^{\mathbb {N} }} with the product topology, where N {\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 [ 0 , 1 ] {\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 [ 0 , 1 ] {\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 R n {\displaystyle \mathbb {R} ^{n}} induced by the Euclidean metric, which is itself induced by the Euclidean norm. Real line − R {\displaystyle \mathbb {R} }
Unit interval − [ 0 , 1 ] {\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 R 3 . {\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 R 2 {\displaystyle \mathbb {R} ^{2}} or ( 0 , 1 ) 2 {\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 [ 0 , 1 ] 2 ∖ Q 2 , {\displaystyle [0,1]^{2}\setminus \mathbb {Q} ^{2},} where [ 0 , 1 ] 2 ∩ Q 2 {\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 − [ 0 , 1 / 1 ] × [ 0 , 1 / 2 ] × [ 0 , 1 / 3 ] × ⋯ {\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 R {\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 x ∈ X {\displaystyle x\in X} be a non-isolated point of X , {\displaystyle X,} let d ∉ X {\displaystyle d\not \in X} be arbitrary, and let Y = X ∪ { d } . {\displaystyle Y=X\cup \{d\}.} Then τ = { V ⊆ Y : either V or ( V ∖ { d } ) ∪ { x } is an open subset of X } {\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 Y {\displaystyle Y} and x {\displaystyle x} and d {\displaystyle d} have the same neighborhood filters in Y . {\displaystyle Y.} In this way, x {\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 X {\displaystyle X} that is homeomorphic to X × X . {\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
References
External links π-Base: An Interactive Encyclopedia of Topological Spaces