View source for List of topologies

{{short description|List of concrete topologies and topological spaces}}
{{Main|List of topology topics}}

The following is a list of named [[Topology (structure)|topologies]] or [[topological space]]s, many of which are counterexamples in [[topology]] and related branches of [[mathematics]]. This is not a list of [[Topological property|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==

{{See also|Cardinality|Ordinal number}}

* [[Cocountable topology]]
** Given a topological space <math>(X, \tau),</math> the ''[[Cocountable extension topology|{{dfn|cocountable extension topology}}]]'' on <math>X</math> is the topology having as a subbasis the union of {{math|τ}} and the family of all subsets of <math>X</math> whose complements in <math>X</math> 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 [[Kolmogorov space|T<sub>0</sub>]]. However, from the viewpoint of [[algebraic topology]], it has the remarkable property that it is indistinguishable from the [[circle]] <math>S^1.</math>
* [[Sierpiński space]], also called the [[connected two-point set]] − A 2-point set <math>\{0, 1\}</math> with the [[particular point topology]] <math>\{\varnothing, \{1\}, \{0,1\}\}.</math>

==Integers==

* [[Arens–Fort space]] − A Hausdorff, regular, normal space that is not first-countable or compact. It has an element (i.e. <math>p := (0, 0)</math>) for which there is no sequence in <math>X \setminus \{p\}</math> that converges to <math>p</math> but there is a sequence <math>x_\bull = \left(x_i\right)_{i=1}^\infty</math> in <math>X \setminus \{(0, 0)\}</math> such that <math>(0, 0)</math> is a cluster point of <math>x_\bull.</math>
* [[Arithmetic progression topologies]]
* [[Baire space (set theory)|The Baire space]] − <math>\N^{\N}</math> with the product topology, where <math>\N</math> 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==

{{See also|List of fractals by Hausdorff dimension|Fractal}}

* [[Apollonian gasket]]
* [[Cantor set]] − A subset of the closed interval <math>[0, 1]</math> with remarkable properties.
** [[Cantor dust]]
** [[Cantor space]]
* [[Koch snowflake]]
* [[Menger sponge]]
* [[Mosely snowflake]]
* [[Sierpiński carpet]]
* [[Sierpiński triangle]]
* [[Smith–Volterra–Cantor set]], also called the {{em|fat Cantor set}} − A closed [[nowhere dense]] (and thus [[Meagre set|meagre]]) subset of the unit interval <math>[0, 1]</math> 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==

{{See also|Preorder|Partially ordered set}}

* [[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 ''{{dfn|Alexandrov double arrow space}}'' and the ''{{dfn|two arrows space}}'' − All compact separable ordered spaces are order-isomorphic to a subset of the split interval. It is [[Compact Hausdorff space|compact Hausdorff]], [[Hereditarily Lindelöf space|hereditarily Lindelöf]], and [[Hereditarily separable space|hereditarily separable]] but not [[metrizable space|metrizable]]. Its metrizable [[Subspace topology|subspace]]s are all countable.
* [[Specialization (pre)order]]

==Manifolds and complexes==

{{See also|Topological manifold|Smooth manifold}}

* [[Branching line]] − A [[non-Hausdorff manifold]].
* [[Double origin topology]]
* [[E8 manifold|E<sub>8</sub> manifold]] − A [[topological manifold]] that does not admit a [[smooth structure]].
* [[Euclidean topology]] − The [[natural topology]] on [[Euclidean space]] <math>\Reals^n</math> induced by the [[Euclidean distance|Euclidean metric]], which is itself induced by the [[Euclidean norm]]. 
** [[Real line]] − <math>\Reals</math>
** [[Unit interval]] − <math>[0, 1]</math>
* [[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 [[Collapse (topology)|collapsible]].
* [[Klein bottle]]
* [[Lens space]]
* [[Line with two origins]], also called the ''{{dfn|bug-eyed line}}'' − It is a [[non-Hausdorff manifold]]. It is [[locally homeomorphic]] to [[Euclidean space]] and thus [[locally metrizable space|locally metrizable]] (but not [[Metrizable space|metrizable]]) and [[Locally Hausdorff space|locally Hausdorff]] (but not [[Hausdorff space|Hausdorff]]). It is also a [[T1 space|T<sub>1</sub>]] [[locally regular space]] but not a [[semiregular space]].
* [[Prüfer manifold]] − A Hausdorff 2-dimensional real analytic manifold that is not [[Paracompact space|paracompact]].
* [[Real projective line]]
* [[Torus]]
** [[3-torus]]
** [[Solid torus]]
* [[Unknot]]
* [[Whitehead manifold]] − An open [[3-manifold]] that is [[contractible]], but not [[homeomorphic]] to <math>\Reals^3.</math>

===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 <math>\Reals^2</math> or <math>(0, 1)^2</math> that all have the same boundary.

===Unique===

* [[Hantzsche–Wendt manifold]] − A compact, orientable, [[Flat manifold|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 [[Topology|point-set topology]]. 

* [[Alexandroff plank]]
* [[Appert topology]] − A Hausdorff, [[perfectly normal space|perfectly normal]] (T<sub>6</sub>), [[zero-dimensional space]] that is countable, but neither [[first countable]], [[Locally compact space|locally compact]], nor [[countably compact]].
* [[Arens square]]
* [[Bullet-riddled square]] - The space <math>[0, 1]^2 \setminus \Q^2,</math> where <math>[0, 1]^2 \cap \Q^2</math> is the set of bullets. Neither of these sets is [[Jordan measure|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]] − <math>[0, 1/1] \times [0, 1/2] \times [0, 1/3] \times \cdots</math> with the [[product topology]].
* [[Infinite broom]]
* [[Integer broom topology]]
* [[K-topology]]
* [[Knaster–Kuratowski fan]]
* [[Long line (topology)]]
* [[Moore plane]], also called the ''{{dfn|Niemytzki plane}}'' − A [[first countable]], [[Separable space|separable]], [[completely regular]], Hausdorff, [[Moore space (topology)|Moore space]] that is not [[normal space|normal]], [[Lindelöf space|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 T<sub>0</sub> but not T<sub>1</sub>.
* [[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 space|metacompact]] nor [[Paracompact space|paracompact]].
* [[Rational sequence topology]]
* [[Sorgenfrey line]], which is <math>\Reals</math> endowed with [[lower limit topology]] − It is Hausdorff, perfectly normal, first-countable, separable, paracompact, Lindelöf, [[Baire space|Baire]], and a [[Moore space (topology)|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 (topology)|Moore space]] that is neither [[normal space|normal]], [[Paracompact space|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 topology|natural topologies]]. 

* [[Adjunction space]]
* [[Disjoint union (topology)]]
* [[Extension topology]]
* [[Initial topology]]
* [[Final topology]]
* [[Product topology]]
* [[Quotient topology]]
* [[Subspace topology]]
* [[Weak topology]]

===Compactifications===

[[Compactification (mathematics)|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]] ([[Modes of convergence (annotated index)|annotated index]])
* [[Operator topologies]]
* [[Pointwise convergence]]
** [[Weak convergence (Hilbert space)]]
** [[Weak* topology]]
* [[Polar topology]]
* [[Strong dual space|Strong dual topology]]
* [[Topologies on spaces of linear maps]]

===Other induced topologies===

* [[Box topology]]
* [[Compact complement topology]]
* <em>[[Duplication of a point]]</em>: Let <math>x \in X</math> be a non-[[isolated point]] of <math>X,</math> let <math>d \not\in X</math> be arbitrary, and let <math>Y = X \cup \{d\}.</math> Then <math>\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>Y</math> and <math>x</math> and <math>d</math> have the same [[neighborhood filter]]s in <math>Y.</math> In this way, <math>x</math> has been duplicated.{{sfn|Wilansky|2008|p=35}}
* [[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]], [[Dimension theory|one-dimensional]] topological space <math>X</math> that is homeomorphic to <math>X \times X.</math>
* [[Half-disk topology]]
* [[Hedgehog space]]
* [[Partition topology]]
* [[Zariski topology]]

==See also==

* {{annotated link|Counterexamples in Topology|''Counterexamples in Topology''}}
* {{annotated link|List of Banach spaces}}
* {{annotated link|List of fractals by Hausdorff dimension}}
* {{annotated link|List of manifolds}}
* {{annotated link|List of topologies on the category of schemes}}
* {{annotated link|List of topology topics}}
* {{annotated link|Lists of mathematics topics}}
* {{annotated link|Natural topology}}
* {{annotated link|Table of Lie groups}}

==Citations==

{{reflist}}
{{reflist|group=note}}

==References==

{{refbegin|2}}
* {{Adams Franzosa Introduction to Topology Pure and Applied}} <!--{{sfn|Adams|Franzosa|2009|p=}}-->
* {{Arkhangel'skii Ponomarev Fundamentals of General Topology Problems and Exercises|edition=2}} <!--{{sfn|Arkhangelʹskiĭ|Ponomarev|1984|p=}}-->
* {{Bourbaki General Topology Part I Chapters 1-4}} <!--{{sfn|Bourbaki|1989|p=}}-->
* {{Bourbaki General Topology Part II Chapters 5-10}} <!--{{sfn|Bourbaki|1989|p=}}-->
* {{Comfort Negrepontis The Theory of Ultrafilters 1974}} <!--{{sfn|Comfort|Negrepontis|1974|p=}}-->
* {{Dixmier General Topology}} <!--{{sfn|Dixmier|1984|p=}}-->
* {{Császár General Topology}} <!--{{sfn|Császár|1978|p=}}-->
* {{Dolecki Mynard Convergence Foundations Of Topology}} <!--{{sfn|Dolecki|Mynard|2016|p=}}-->
* {{Dugundji Topology}} <!--{{sfn|Dugundji|1966|p=}}-->
* {{Howes Modern Analysis and Topology 1995}} <!--{{sfn|Howes|1995|p=}}-->
* {{Jarchow Locally Convex Spaces}} <!--{{sfn|Jarchow|1981|p=}}-->
* {{Joshi Introduction to General Topology}} <!--{{sfn|Joshi|1983|p=}}-->
* {{Kelley General Topology}} <!--{{sfn|Kelley|1975|p=}}-->
* {{Köthe Topological Vector Spaces I}} <!--{{sfn|Köthe|1969|p=}}-->
* {{Munkres Topology|edition=2}} <!--{{sfn|Munkres|2000|p=}}-->
* {{Schechter Handbook of Analysis and Its Foundations}} <!--{{sfn|Schechter|1996|p=}}-->
* {{Schubert Topology}} <!--{{sfn|Schubert|1968|p=}}-->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!--{{sfn|Wilansky|2013|p=}}-->
* {{Wilansky Topology for Analysis 2008}} <!--{{sfn|Wilansky|2008|p=}}-->
* {{Willard General Topology}} <!--{{sfn|Willard|2004|p=}}-->
{{refend}}

==External links==

* [http://topology.jdabbs.com π-Base: An Interactive Encyclopedia of Topological Spaces]

{{Topology}}

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