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In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a topological property or topological invariant is a property of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
that is
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
under
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
s. Alternatively, a topological property is a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space ''X'' possesses that property every space homeomorphic to ''X'' possesses that property. Informally, a topological property is a property of the space that can be expressed using
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s. A common problem in topology is to decide whether two topological spaces are
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
or not. To prove that two spaces are ''not'' homeomorphic, it is sufficient to find a topological property which is not shared by them.


Properties of topological properties

A property P is: * Hereditary, if for every topological space (X, \mathcal) and X' \subset X, the subspace (X', \mathcal, X') has property P. * Weakly hereditary, if for every topological space (X, \mathcal) and closed X' \subset X, the subspace (X', \mathcal, X') has property P.


Common topological properties


Cardinal functions

* The
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
, ''X'', of the space ''X''. * The cardinality \vert''τ''(''X'')\vert of the topology (the set of open subsets) of the space ''X''. * ''Weight'' ''w''(''X''), the least cardinality of a basis of the topology of the space ''X''. * ''Density'' ''d''(''X''), the least cardinality of a subset of ''X'' whose closure is ''X''.


Separation

Note that some of these terms are defined differently in older mathematical literature; see
history of the separation axioms The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept. Origins Before the current general definition of topological space, th ...
. * T0 or Kolmogorov. A space is
Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
if for every pair of distinct points ''x'' and ''y'' in the space, there is at least either an open set containing ''x'' but not ''y'', or an open set containing ''y'' but not ''x''. * T1 or Fréchet. A space is Fréchet if for every pair of distinct points ''x'' and ''y'' in the space, there is an open set containing ''x'' but not ''y''. (Compare with T0; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T1 if all its singletons are closed. T1 spaces are always T0. * Sober. A space is
sober In cryptography, SOBER is a family of stream ciphers initially designed by Greg Rose of QUALCOMM Australia starting in 1997. The name is a contrived acronym for ''S''eventeen ''O''ctet ''B''yte ''E''nabled ''R''egister. Initially the cipher wa ...
if every irreducible closed set ''C'' has a unique generic point ''p''. In other words, if ''C'' is not the (possibly nondisjoint) union of two smaller closed subsets, then there is a ''p'' such that the closure of equals ''C'', and ''p'' is the only point with this property. * T2 or Hausdorff. A space is Hausdorff if every two distinct points have disjoint neighbourhoods. T2 spaces are always T1. * T or Urysohn. A space is
Urysohn Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are f ...
if every two distinct points have disjoint ''closed'' neighbourhoods. T spaces are always T2. * Completely T2 or completely Hausdorff. A space is completely T2 if every two distinct points are
separated by a function In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
. Every completely Hausdorff space is Urysohn. * Regular. A space is regular if whenever ''C'' is a closed set and ''p'' is a point not in ''C'', then ''C'' and ''p'' have disjoint neighbourhoods. * T3 or Regular Hausdorff. A space is regular Hausdorff if it is a regular T0 space. (A regular space is Hausdorff if and only if it is T0, so the terminology is
consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
.) * Completely regular. A space is
completely regular In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
if whenever ''C'' is a closed set and ''p'' is a point not in ''C'', then ''C'' and are
separated by a function In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
. * T, Tychonoff, Completely regular Hausdorff or Completely T3. A
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
is a completely regular T0 space. (A completely regular space is Hausdorff if and only if it is T0, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff. * Normal. A space is
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit
partitions of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0, ...
. * T4 or Normal Hausdorff. A normal space is Hausdorff if and only if it is T1. Normal Hausdorff spaces are always Tychonoff. * Completely normal. A space is
completely normal In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. Th ...
if any two
separated sets In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
have disjoint neighbourhoods. * T5 or Completely normal Hausdorff. A completely normal space is Hausdorff if and only if it is T1. Completely normal Hausdorff spaces are always normal Hausdorff. * Perfectly normal. A space is
perfectly normal ''Perfectly Normal'' is a Canadian comedy film directed by Yves Simoneau, which premiered at the 1990 Festival of Festivals, before going into general theatrical release in 1991. Simoneau's first English-language film, it was written by Eugene Lip ...
if any two disjoint closed sets are
precisely separated by a function In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
. A perfectly normal space must also be completely normal. * T6 or Perfectly normal Hausdorff, or perfectly T4. A space is perfectly normal Hausdorff, if it is both perfectly normal and T1. A perfectly normal Hausdorff space must also be completely normal Hausdorff. * Discrete space. A space is
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...
if all of its points are completely isolated, i.e. if any subset is open. * Number of isolated points. The number of
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...
s of a topological space.


Countability conditions

* Separable. A space is separable (topology), separable if it has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
dense subset. * First-countable. A space is
first-countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
if every point has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
local base. * Second-countable. A space is
second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
if it has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.


Connectedness

* Connected. A space is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
if it is not the union of a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only
clopen set In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
s are the empty set and itself. * Locally connected. A space is
locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness a ...
if every point has a local base consisting of connected sets. * Totally disconnected. A space is
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
if it has no connected subset with more than one point. * Path-connected. A space ''X'' is
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
if for every two points ''x'', ''y'' in ''X'', there is a path ''p'' from ''x'' to ''y'', i.e., a continuous map ''p'':  ,1nbsp;→ ''X'' with ''p''(0) = ''x'' and ''p''(1) = ''y''. Path-connected spaces are always connected. * Locally path-connected. A space is
locally path-connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness a ...
if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected. * Arc-connected. A space ''X'' is arc-connected if for every two points ''x'', ''y'' in ''X'', there is an arc ''f'' from ''x'' to ''y'', i.e., an
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
continuous map ''f'':  ,1nbsp;→ ''X'' with ''p''(0) = ''x'' and ''p''(1) = ''y''. Arc-connected spaces are path-connected. * Simply connected. A space ''X'' is
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
if it is path-connected and every continuous map ''f'': S1 → ''X'' is
homotopic In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to a constant map. *Locally simply connected. A space ''X'' is
locally simply connected In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected. The circle is an example of a locally ...
if every point ''x'' in ''X'' has a local base of neighborhoods ''U'' that is simply connected. *Semi-locally simply connected. A space ''X'' is
semi-locally simply connected In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space ''X'' is semi-locally simply connected if ...
if every point has a local base of neighborhoods ''U'' such that ''every'' loop in ''U'' is contractible in ''X''. Semi-local simple connectivity, a strictly weaker condition than local simple connectivity, is a necessary condition for the existence of a
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
. * Contractible. A space ''X'' is
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...
if the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on ''X'' is homotopic to a constant map. Contractible spaces are always simply connected. * Hyperconnected. A space is
hyperconnected In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
if no two non-empty open sets are disjoint. Every hyperconnected space is connected. * Ultraconnected. A space is
ultraconnected In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint.PlanetMath Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersecti ...
if no two non-empty closed sets are disjoint. Every ultraconnected space is path-connected. * Indiscrete or trivial. A space is indiscrete if the only open sets are the empty set and itself. Such a space is said to have the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...
.


Compactness

* Compact. A space is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
if every
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
has a finite ''subcover''. Some authors call these spaces quasicompact and reserve compact for Hausdorff spaces where every open cover has finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal. * Sequentially compact. A space is
sequentially compact In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X. Every metric space is naturally a topological space, and for metric spaces, the notio ...
if every sequence has a convergent subsequence. * Countably compact. A space is
countably compact In mathematics a topological space is called countably compact if every countable open cover has a finite subcover. Equivalent definitions A topological space ''X'' is called countably compact if it satisfies any of the following equivalent condit ...
if every countable open cover has a finite subcover. * Pseudocompact. A space is
pseudocompact In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of ps ...
if every continuous real-valued function on the space is bounded. * σ-compact. A space is σ-compact if it is the union of countably many compact subsets. * Lindelöf. A space is Lindelöf if every open cover has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
subcover. * Paracompact. A space is
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal. * Locally compact. A space is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
if every point has a local base consisting of compact neighbourhoods. Slightly different definitions are also used. Locally compact Hausdorff spaces are always Tychonoff. * Ultraconnected compact. In an ultra-connected compact space ''X'' every open cover must contain ''X'' itself. Non-empty ultra-connected compact spaces have a largest proper open subset called a monolith.


Metrizability

* Metrizable. A space is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
if it is homeomorphic to a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable. Moreover, a topological space (X,T) is said to be metrizable if there exists a metric for X such that the metric topology T(d) is identical with the topology T. * Polish. A space is called
Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, w ...
if it is metrizable with a separable and complete metric. * Locally metrizable. A space is locally metrizable if every point has a metrizable neighbourhood.


Miscellaneous

* Baire space. A space ''X'' is a
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
if it is not meagre in itself. Equivalently, ''X'' is a Baire space if the intersection of countably many dense open sets is dense. * Door space. A topological space is a door space if every subset is open or closed (or both). * Topological Homogeneity. A space ''X'' is (topologically)
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
if for every ''x'' and ''y'' in ''X'' there is a homeomorphism f : X \to X such that f(x) = y. Intuitively speaking, this means that the space looks the same at every point. All
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
s are homogeneous. * Finitely generated or Alexandrov. A space ''X'' is Alexandrov if arbitrary intersections of open sets in ''X'' are open, or equivalently if arbitrary unions of closed sets are closed. These are precisely the finitely generated members of the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again contin ...
and continuous maps. * Zero-dimensional. A space is
zero-dimensional In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical ...
if it has a base of clopen sets. These are precisely the spaces with a small
inductive dimension In the mathematical field of topology, the inductive dimension of a topological space ''X'' is either of two values, the small inductive dimension ind(''X'') or the large inductive dimension Ind(''X''). These are based on the observation that, in ...
of ''0''. * Almost discrete. A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces. * Boolean. A space is Boolean if it is zero-dimensional, compact and Hausdorff (equivalently, totally disconnected, compact and Hausdorff). These are precisely the spaces that are homeomorphic to the
Stone space In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in ...
s of
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
s. * Reidemeister torsion * \kappa-resolvable. A space is said to be κ-resolvable (respectively: almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (respectively: almost disjoint over the ideal of nowhere dense subsets). If the space is not \kappa-resolvable then it is called \kappa-irresolvable. * Maximally resolvable. Space X is maximally resolvable if it is \Delta(X)-resolvable, where \Delta(X) = \min\. Number \Delta(X) is called dispersion character of X. * Strongly discrete. Set D is strongly discrete subset of the space X if the points in D may be separated by pairwise disjoint neighborhoods. Space X is said to be strongly discrete if every non-isolated point of X is the accumulation point of some strongly discrete set.


Non-topological properties

There are many examples of properties of
metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, etc, which are not topological properties. To show a property P is not topological, it is sufficient to find two homeomorphic topological spaces X \cong Y such that X has P, but Y does not have P. For example, the metric space properties of boundedness and completeness are not topological properties. Let X = \R and Y = (-\tfrac,\tfrac) be metric spaces with the standard metric. Then, X \cong Y via the homeomorphism \operatorname\colon X \to Y. However, X is complete but not bounded, while Y is bounded but not complete.


See also

*
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
*
Winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turn ...
*
Characteristic class In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
* Characteristic numbers *
Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
*
Knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...
*
Linking number In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In E ...
*
Fixed-point property A mathematical object ''X'' has the fixed-point property if every suitably well-behaved mapping from ''X'' to itself has a fixed point. The term is most commonly used to describe topological spaces on which every continuous mapping has a fixed poi ...
*
Topological quantum number In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are ...
*
Homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
and
Cohomotopy group In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and basepoint-preserving continuous maps to the category of sets and functions. They are d ...
*
Homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
and
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
*
Quantum invariant In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement. List of invariants *Finite type invariant * Kon ...


Citations


References

* * {{refend Simon Moulieras, Maciej Lewenstein and Graciana Puentes, Entanglement engineering and topological protection by discrete-time quantum walks, Journal of Physics B: Atomic, Molecular and Optical Physics 46 (10), 104005 (2013). https://iopscience.iop.org/article/10.1088/0953-4075/46/10/104005/pdf Homeomorphisms ru:Топологический инвариант