List of topologies

• Discrete topology − All subsets are open.
• Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open.

{{See also|Cardinality|Ordinal number}}

• Cocountable topology
• Given a topological space $(X,\; \backslash tau),$ the Cocountable topology#Cocountable extension topology on $X$ is the topology having as a subbasis the union of {{math|τ}} and the family of all subsets of $X$ whose complements in $X$ are countable.
• Cofinite topology
• Double-pointed cofinite topology
• Ordinal number topology
• Pseudo-arc
• Ran space
• Tychonoff plank

• 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 $S^1.$
• Sierpiński space, also called the connected two-point set − A 2-point set $\backslash \{0,\; 1\backslash \}$ with the particular point topology $\backslash \{\backslash varnothing,\; \backslash \{1\backslash \},\; \backslash \{0,1\backslash \}\backslash \}.$

• Arens–Fort space − A Hausdorff, regular, normal space that is not first-countable or compact. It has an element (i.e. $p\; :=\; (0,\; 0)$) for which there is no sequence in $X\; \backslash setminus\; \backslash \{p\backslash \}$ that converges to $p$ but there is a sequence $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\backslash infty$ in $X\; \backslash setminus\; \backslash \{(0,\; 0)\backslash \}$ such that $(0,\; 0)$ is a cluster point of $x\_\backslash bull.$
• Arithmetic progression topologies
• The Baire space − $\backslash N^\{\backslash N\}$ with the product topology, where $\backslash N$ 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

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

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

{{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, hereditarily Lindelöf, and hereditarily separable but not metrizable. Its metrizable subspaces are all countable.
• Specialization (pre)order

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

• 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 $\backslash Reals^n$ induced by the Euclidean metric, which is itself induced by the Euclidean norm.
• Real line − $\backslash Reals$
• Unit interval − $[0,\; 1]$
• 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 {{dfn|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 $\backslash Reals^3.$

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

• Lakes of Wada − Three disjoint connected open sets of $\backslash Reals^2$ or $(0,\; 1)^2$ that all have the same boundary.

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

• Dogbone space
• Dunce hat (topology)
• Hawaiian earring
• Long line (topology)
• Rose (topology)

• 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

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 $[0,\; 1]^2\; \backslash setminus\; \backslash Q^2,$ where $[0,\; 1]^2\; \backslash cap\; \backslash Q^2$ 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 − $[0,\; 1/1]\; \backslash times\; [0,\; 1/2]\; \backslash times\; [0,\; 1/3]\; \backslash times\; \backslash cdots$ 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, 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 $\backslash Reals$ 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

List of natural topologies.

• Adjunction space
• Disjoint union (topology)
• Extension topology
• Initial topology
• Final topology
• Product topology
• Quotient topology
• Subspace topology
• Weak topology

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

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

• Box topology
• Compact complement topology
• Duplication of a point: Let $x\; \backslash in\; X$ be a non-isolated point of $X,$ let $d\; \backslash not\backslash in\; X$ be arbitrary, and let $Y\; =\; X\; \backslash cup\; \backslash \{d\backslash \}.$ Then $\backslash tau\; =\; \backslash \{V\; \backslash subseteq\; Y\; :\; \backslash text\{\; either\; \}\; V\; \backslash text\{\; or\; \}\; (\; V\; \backslash setminus\; \backslash \{d\backslash \})\; \backslash cup\; \backslash \{x\backslash \}\; \backslash text\{\; is\; an\; open\; subset\; of\; \}\; X\backslash \}$ is a topology on $Y$ and $x$ and $d$ have the same neighborhood filters in $Y.$ In this way, $x$ has been duplicated.{{sfn|Wilansky|2008|p=35}}
• Extension topology

• Auxiliary normed spaces
• Finest locally convex topology
• Finest vector topology
• Helly space
• Mackey topology
• Polar topology
• Vague topology

• 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

• Inductive tensor product
• Injective tensor product
• Projective tensor product
• Tensor product of Hilbert spaces
• Topological tensor product

• Émery topology

• Erdős space − A Hausdorff, totally disconnected, one-dimensional topological space $X$ that is homeomorphic to $X\; \backslash times\; X.$
• Half-disk topology
• Hedgehog space
• Partition topology
• Zariski topology

• {{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}}

{{reflist}}

{{reflist|group=note}}

{{refbegin|2}}

• {{Adams Franzosa Introduction to Topology Pure and Applied}}
• {{Arkhangel'skii Ponomarev Fundamentals of General Topology Problems and Exercises|edition=2}}
• {{Bourbaki General Topology Part I Chapters 1-4}}
• {{Bourbaki General Topology Part II Chapters 5-10}}
• {{Comfort Negrepontis The Theory of Ultrafilters 1974}}
• {{Dixmier General Topology}}
• {{Császár General Topology}}
• {{Dolecki Mynard Convergence Foundations Of Topology}}
• {{Dugundji Topology}}
• {{Howes Modern Analysis and Topology 1995}}
• {{Jarchow Locally Convex Spaces}}
• {{Joshi Introduction to General Topology}}
• {{Kelley General Topology}}
• {{Köthe Topological Vector Spaces I}}
• {{Munkres Topology|edition=2}}
• {{Schechter Handbook of Analysis and Its Foundations}}
• {{Schubert Topology}}
• {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}
• {{Wilansky Topology for Analysis 2008}}
• {{Willard General Topology}}

{{refend}}

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

{{Topology}}

{{DEFAULTSORT:Topologies}}

Category:General topology

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