THC Science

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[edit]

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[edit]

Cocountable topology
- Given a topological space $(X,\tau ),$ 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
Double-pointed cofinite topology
Ordinal number topology
Pseudo-arc
Ran space
Tychonoff plank

Finite spaces[edit]

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 $\{0,1\}$ with the particular point topology $\{\varnothing ,\{1\},\{0,1\}\}.$

Integers[edit]

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 $X\setminus \{p\}$ that converges to $p$ but there is a sequence $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ in $X\setminus \{(0,0)\}$ such that $(0,0)$ is a cluster point of $x_{\bullet }.$
Arithmetic progression topologies
The Baire space − $\mathbb {N} ^{\mathbb {N} }$ with the product topology, where $\mathbb {N}$ 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[edit]

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 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.

Orders[edit]

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[edit]

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 $\mathbb {R} ^{n}$ induced by the Euclidean metric, which is itself induced by the Euclidean norm.
- Real line − $\mathbb {R}$
- 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 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 $\mathbb {R} ^{3}.$

Hyperbolic geometry[edit]

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

Paradoxical spaces[edit]

Lakes of Wada − Three disjoint connected open sets of $\mathbb {R} ^{2}$ or $(0,1)^{2}$ that they all have the same boundary.

Unique[edit]

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[edit]

Embeddings and maps between spaces[edit]

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)[edit]

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 $[0,1]^{2}\setminus \mathbb {Q} ^{2},$ where $[0,1]^{2}\cap \mathbb {Q} ^{2}$ 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 − $[0,1/1]\times [0,1/2]\times [0,1/3]\times \cdots$ 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 $\mathbb {R}$ 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[edit]

Natural topologies[edit]

List of natural topologies.

Compactifications[edit]

Compactifications include:

Topologies of uniform convergence[edit]

This lists named topologies of uniform convergence.

Other induced topologies[edit]

Box topology
Compact complement topology
Duplication of a point: Let $x\in X$ be a non-isolated point of $X,$ let $d\not \in X$ be arbitrary, and let $Y=X\cup \{d\}.$ Then $\tau =\{V\subseteq Y:{\text{ either }}V{\text{ or }}(V\setminus \{d\})\cup \{x\}{\text{ is an open subset of }}X\}$ is a topology on $Y$ and $x$ and $d$ have the same neighborhood filters in $Y.$ In this way, $x$ has been duplicated.^[1]
Extension topology

Functional analysis[edit]

Operator topologies[edit]

Tensor products[edit]

Probability[edit]

Émery topology

Other topologies[edit]

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

Citations[edit]

^ Wilansky 2008, p. 35.

References[edit]

Adams, Colin; Franzosa, Robert (2009). Introduction to Topology: Pure and Applied. New Delhi: Pearson Education. ISBN 978-81-317-2692-1. OCLC 789880519.
Arkhangel'skii, Alexander Vladimirovich; Ponomarev, V.I. (1984). Fundamentals of General Topology: Problems and Exercises. Mathematics and Its Applications. Vol. 13. Dordrecht Boston: D. Reidel. ISBN 978-90-277-1355-1. OCLC 9944489.
Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
Bourbaki, Nicolas (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. Éléments de mathématique. Vol. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063.
Comfort, William Wistar; Negrepontis, Stylianos (1974). The Theory of Ultrafilters. Vol. 211. Berlin Heidelberg New York: Springer-Verlag. ISBN 978-0-387-06604-2. OCLC 1205452.
Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303.
Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011.
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.
Howes, Norman R. (23 June 1995). Modern Analysis and Topology. Graduate Texts in Mathematics. New York: Springer-Verlag Science & Business Media. ISBN 978-0-387-97986-1. OCLC 31969970. OL 1272666M.
Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
Joshi, K. D. (1983). Introduction to General Topology. New York: John Wiley and Sons Ltd. ISBN 978-0-85226-444-7. OCLC 9218750.
Kelley, John L. (1975). General Topology. Graduate Texts in Mathematics. Vol. 27. New York: Springer Science & Business Media. ISBN 978-0-387-90125-1. OCLC 338047.
Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. 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.
Schubert, Horst (1968). Topology. London: Macdonald & Co. ISBN 978-0-356-02077-8. OCLC 463753.
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
Wilansky, Albert (17 October 2008) [1970]. Topology for Analysis. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-46903-4. OCLC 227923899.
Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.

External links[edit]

π-Base: An Interactive Encyclopedia of Topological Spaces

[FOOTNOTEWilansky200835-1] Wilansky 2008, p. 35.

[1]

Topology
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Metrics and properties	Euler characteristic Betti number Winding number Chern number Orientability
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