Short description: List of concrete topologies and topological spaces

Main page: List of topology topics

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.

## Contents

* 1 Widely known topologies
* 2 Counter-example topologies
* 2.1 Hyperbolic geometry
* 2.2 Paradoxical spaces
* 2.3 Pathological embeddings of spaces
* 2.4 Unique
* 3 Topologies defined in terms of other topologies
* 3.1 Natural topologies
* 3.2 Compactifications
* 3.3 Topologies of uniform convergence
* 3.4 Functional analysis
* 3.4.1 Operator topologies
* 3.4.2 Tensor products
* 3.5 Other induced topologies
* 4 Fractal spaces
* 5 Topologies related to other structures
* 6 Other topologies
* 7 See also
* 8 Citations
* 9 References

## Widely known topologies

* The Baire space − [math]\displaystyle{ \N^{\N} }[/math] with the product topology, where [math]\displaystyle{ \N }[/math] denotes the natural numbers endowed with the discrete topology. It is the space of all sequences of natural numbers.
* Cantor set − A subset of the closed interval [math]\displaystyle{ [0, 1] }[/math] with remarkable properties.
* Cantor dust
* Discrete topology − All subsets are open.
* Euclidean topology − The natural topology on Euclidean space [math]\displaystyle{ \R^n }[/math] induced by the Euclidean metric, which is itself induced by the Euclidean norm.
* Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open.
* Klein bottle
* Real projective line
* Torus
* 3-torus
* Solid torus

## Counter-example topologies

The following topologies are a known source of counterexamples for point-set topology.

* Appert topology − A Hausdorff, perfectly normal (T6), zero-dimensional space that is countable, but neither first countable, locally compact, nor countably compact.
* Arens–Fort space − A Hausdorff, regular, normal space that is not first-countable or compact. It has an element (i.e. [math]\displaystyle{ p := (0, 0) }[/math]) for which there is no sequence in [math]\displaystyle{ X \setminus \\{ p \\} }[/math] that converges to [math]\displaystyle{ p }[/math] but there is a sequence [math]\displaystyle{ x_{\bull} = \left( x_i \right)_{i=1}^{\infty} }[/math] in [math]\displaystyle{ X \setminus \\{ (0, 0) \\} }[/math] such that [math]\displaystyle{ (0, 0) }[/math] is a cluster point of [math]\displaystyle{ x_{\bull}. }[/math]
* Branching line − A non-Hausdorff manifold.
* Bullet-riddled square - The space [math]\displaystyle{ [0, 1]^2 \setminus \Q^2, }[/math] where [math]\displaystyle{ [0, 1]^2 \cap \Q^2 }[/math] is the set of bullets. Neither of these sets is Jordan measurable although both are Lebesgue measurable.
* Comb space
* Dogbone space
* Dunce hat (topology)
* E8 manifold − A topological manifold that does not admit a smooth structure.
* Excluded point topology − A topological space where the open sets are defined in terms of the exclusion of a particular point.
* Fort space
* House with two rooms − A contractible, 2-dimensional Simplicial complex that is not collapsible.
* Infinite broom
* Integer broom topology
* K-topology
* Lens space
* Lexicographic order topology on the unit square
* Line with two origins, also called the  bug-eyed line − It is a non-Hausdorff manifold and a locally regular space but not a semiregular space.
* 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.
* 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.
* Prüfer manifold − A Hausdorff 2-dimensional real analytic manifold that is not paracompact.
* Smith–Volterra–Cantor set − A closed nowhere dense (and thus meagre) subset of the unit interval [math]\displaystyle{ [0, 1] }[/math] that has positive Lebesgue measure.
* Sorgenfrey line, which is [math]\displaystyle{ \R }[/math] 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
* Whitehead manifold − An open 3-manifold that is contractible, but not homeomorphic to [math]\displaystyle{ \R^3. }[/math]

### Hyperbolic geometry

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

### Paradoxical spaces

* Gabriel's horn − It has infinite surface area but finite volume.

### Pathological embeddings of 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.

### Unique

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

## Topologies defined in terms of other topologies

### Natural topologies

List of natural topologies.

* Corona set
* Disjoint union (topology)
* Extension topology
* Initial topology
* Final topology
* Product topology
* Quotient topology
* Subspace topology
* Weak topology

### Compactifications

* Alexandroff extension
* Projectively extended real line
* Bohr compactification
* Eells–Kuiper manifold
* Stone–Čech compactification
* Stone topology
* 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

### Functional analysis

* Auxiliary normed spaces
* Finest locally convex topology
* Finest vector topology
* Mackey topology
* Polar topology

#### Operator topologies

* Dual topology
* Norm topology
* Operator topologies
* Pointwise convergence
* Weak convergence (Hilbert space)
* Weak* 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

### Other induced topologies

* Box topology
* Duplication of a point: Let [math]\displaystyle{ x \in X }[/math] be a non-isolated point of [math]\displaystyle{ X, }[/math] let [math]\displaystyle{ d \not\in X }[/math] be arbitrary, and let [math]\displaystyle{ Y = X \cup \\{ d \\}. }[/math] Then [math]\displaystyle{ \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]\displaystyle{ Y }[/math] and x and d have the same neighborhood filters in [math]\displaystyle{ Y. }[/math] In this way, x has been duplicated.[1]

## Fractal spaces

* Apollonian gasket
* Cantor set
* Koch snowflake
* Menger sponge
* Mosely snowflake
* Sierpiński carpet
* Sierpiński triangle

## Topologies related to other structures

* Order topology

## Other topologies

* Cantor space
* Cocountable topology
* Given a topological space [math]\displaystyle{ (X, \tau), }[/math] the  cocountable extension topology on X is the topology having as a subbasis the union of τ and the family of all subsets of X whose complements in X are countable.
* Cofinite topology
* Discrete two-point space − The simplest example of a totally disconnected discrete space.
* Double-pointed cofinite topology
* Erdős space − A Hausdorff, totally disconnected, one-dimensional topological space [math]\displaystyle{ X }[/math] that is homeomorphic to [math]\displaystyle{ X \times X. }[/math]
* Fake 4-ball − A compact contractible topological 4-manifold.
* Half-disk topology
* Hawaiian earring
* Hedgehog space
* Long line (topology)
* 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 [math]\displaystyle{ S^1. }[/math]
* Rose (topology)
* 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.
* Zariski topology

## See also

* Counterexamples in Topology - Book by Lynn Steen
* List of Banach spaces
* List of manifolds
* List of topology topics
* Lists of mathematics topics - None
* Natural topology - Notion in topology

## Citations

1. ↑ Wilansky 2008, p. 35.

## References

* Template:Adams Franzosa Introduction to Topology Pure and Applied
* Template:Arkhangel'skii Ponomarev Fundamentals of General Topology Problems and Exercises
* Bourbaki, Nicolas (1989). General Topology: Chapters 1–4. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129. https://doku.pub/documents/31425779-nicolas-bourbaki-general-topology-part-i1pdf-30j71z37920w.
* Template:Bourbaki General Topology Part II Chapters 5-10
* Template:Comfort Negrepontis The Theory of Ultrafilters 1974
* Template:Dixmier General Topology
* Template:Császár General Topology
* 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.
* Template:Howes Modern Analysis and Topology 1995
* Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
* Template:Joshi Introduction to General Topology
* Template:Kelley General Topology
* Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. OCLC 840293704.
* Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
* Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
* Template:Schubert Topology
* Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
* Template:Wilansky Topology for Analysis 2008
* Willard, Stephen (2004). General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.

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