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Topological space

From Wikipedia, the free encyclopedia

Mathematical space with a notion of closeness

Inmathematics, atopological space is, roughly speaking, ageometrical space in whichcloseness is defined but cannot necessarily be measured by a numericdistance. More specifically, a topological space is aset whose elements are calledpoints, along with an additional structure called a topology, which can be defined as a set ofneighbourhoods for each point that satisfy someaxioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition throughopen sets.

A topological space is the most general type of amathematical space that allows for the definition oflimits,continuity, andconnectedness.^[1]^[2] Common types of topological spaces includeEuclidean spaces,metric spaces andmanifolds.

Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is calledgeneral topology (or point-set topology).

History

A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitesimal distance from A are deflected infinitesimally from one and the same plane passing through A.

Carl Friedrich Gauss, Gauss 1827

Around 1735,Leonhard Euler discovered theformula $V-E+F=2$ relating the number of vertices (V), edges (E) and faces (F) of aconvex polyhedron, and hence of aplanar graph. The study and generalization of this formula, specifically by Cauchy (1789–1857) and L'Huilier (1750–1840), boosted the study of topology. In 1827,Carl Friedrich Gauss publishedGeneral investigations of curved surfaces, which in section 3 defines the curved surface in a similar manner to the modern topological understanding.

Yet, "until Riemann's work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered".^[3] " Möbius and Jordan seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are homeomorphic or not."^[3]

The subject is clearly defined byFelix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced byJohann Benedict Listing in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created byHenri Poincaré. His first article on this topic appeared in 1894.^[4] In the 1930s,James Waddell Alexander II andHassler Whitney first expressed the idea that a surface is a topological space that islocally like a Euclidean plane.

Topological spaces were first defined byFelix Hausdorff in 1914 in his seminal "Principles of Set Theory".Metric spaces had been defined earlier in 1906 byMaurice Fréchet, though it was Hausdorff who popularised the term "metric space" (German:metrischer Raum).^[5]^[6]^{[better source needed]}

Definitions

Main article:Axiomatic foundations of topological spaces

The utility of the concept of atopology is shown by the fact that there are several equivalent definitions of thismathematical structure. Thus one chooses theaxiomatization suited for the application. The most commonly used is that in terms ofopen sets, but perhaps more intuitive is that in terms ofneighbourhoods and so this is given first.

Definition via neighbourhoods

This axiomatization is due toFelix Hausdorff.Let $X {\displaystyle X}$ be a (possibly empty) set. The elements of $X {\displaystyle X}$ are usually calledpoints, though they can be any mathematical object. Let ${\mathcal {N}}$ be afunction assigning to each $x {\displaystyle x}$ (point) in $X {\displaystyle X}$ a non-empty collection ${\mathcal {N}}(x)$ of subsets of $X . {\displaystyle X.}$ The elements of ${\mathcal {N}}(x)$ will be calledneighbourhoods of $x {\displaystyle x}$ with respect to ${\mathcal {N}}$ (or, simply,neighbourhoods of $x {\displaystyle x}$ ). The function ${\mathcal {N}}$ is called aneighbourhood topology if theaxioms below^[7] are satisfied; and then $X {\displaystyle X}$ with ${\mathcal {N}}$ is called atopological space.

If $N {\displaystyle N}$ is a neighbourhood of $x {\displaystyle x}$ (i.e., $N\in {\mathcal {N}}(x)$ ), then $x\in N.$ In other words, each point of the set $X {\displaystyle X}$ belongs to every one of its neighbourhoods with respect to ${\mathcal {N}}$ .
If $N {\displaystyle N}$ is a subset of $X {\displaystyle X}$ and includes a neighbourhood of $x, {\displaystyle x,}$ then $N {\displaystyle N}$ is a neighbourhood of $x . {\displaystyle x.}$ I.e., everysuperset of a neighbourhood of a point $x\in X$ is again a neighbourhood of $x . {\displaystyle x.}$
Theintersection of two neighbourhoods of $x {\displaystyle x}$ is a neighbourhood of $x . {\displaystyle x.}$
Any neighbourhood $N {\displaystyle N}$ of $x {\displaystyle x}$ includes a neighbourhood $M {\displaystyle M}$ of $x {\displaystyle x}$ such that $N {\displaystyle N}$ is a neighbourhood of each point of $M . {\displaystyle M.}$

The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of $X . {\displaystyle X.}$

A standard example of such a system of neighbourhoods is for the real line $\mathbb {R} ,$ where a subset $N {\displaystyle N}$ of $\mathbb {R}$ is defined to be aneighbourhood of a real number $x {\displaystyle x}$ if it includes an open interval containing $x . {\displaystyle x.}$

Given such a structure, a subset $U {\displaystyle U}$ of $X {\displaystyle X}$ is defined to beopen if $U {\displaystyle U}$ is a neighbourhood of all points in $U . {\displaystyle U.}$ The open sets then satisfy the axioms given below in the next definition of a topological space. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining $N {\displaystyle N}$ to be a neighbourhood of $x {\displaystyle x}$ if $N {\displaystyle N}$ includes an open set $U {\displaystyle U}$ such that $x\in U.$ ^[8]

Definition via open sets

Atopology on asetX may be defined as a collection $\tau$ ofsubsets ofX, calledopen sets and satisfying the following axioms:^[9]

Theempty set and $X {\displaystyle X}$ itself belong to $\tau .$
Any arbitrary (finite or infinite)union of members of $\tau$ belongs to $\tau .$
The intersection of any finite number of members of $\tau$ belongs to $\tau .$

As this definition of a topology is the most commonly used, the set $\tau$ of the open sets is commonly called atopology on $X . {\displaystyle X.}$

A subset $C\subseteq X$ is said to beclosed in $(X,\tau )$ if itscomplement $X\setminus C$ is an open set. Note that from this definition, it follows that the empty set and $X {\displaystyle X}$ are simultaneously openand closed – that is, the two sets are complements of one another, while each of them is, itself, open. In general, any subset of $X {\displaystyle X}$ with this property is said to beclopen.

Examples of topologies

Let $\tau$ be denoted with the circles, here are four examples and two non-examples of topologies on the three-point set $\{1,2,3\}.$ The bottom-left example is not a topology because the union of $\{2\}$ and $\{3\}$ [i.e. $\{2,3\}$ ] is missing; the bottom-right example is not a topology because the intersection of $\{1,2\}$ and $\{2,3\}$ [i.e. $\{2\}$ ], is missing.

{\displaystyle \tau } — Let $\tau$ be denoted with the circles, here are four examples and two non-examples of topologies on the three-point set $\{1,2,3\}.$ The bottom-left example is not a topology because the union of $\{2\}$ and $\{3\}$ [i.e. $\{2,3\}$ ] is missing; the bottom-right example is not a topology because the intersection of $\{1,2\}$ and $\{2,3\}$ [i.e. $\{2\}$ ], is missing.

Given $X=\{1,2,3,4\},$ thetrivial orindiscrete topology on $X {\displaystyle X}$ is thefamily $\tau =\{\{\},\{1,2,3,4\}\}=\{\varnothing ,X\}$ consisting of only the two subsets of $X {\displaystyle X}$ required by the axioms forms a topology on $X . {\displaystyle X.}$
Given $X=\{1,2,3,4\},$ the family $\tau =\{\varnothing ,\{2\},\{1,2\},\{2,3\},\{1,2,3\},X\}$ of six subsets of $X {\displaystyle X}$ forms another topology of $X . {\displaystyle X.}$
Given $X=\{1,2,3,4\},$ thediscrete topology on $X {\displaystyle X}$ is thepower set of $X, {\displaystyle X,}$ which is the family $\tau =\wp (X)$ consisting of all possible subsets of $X . {\displaystyle X.}$ In this case the topological space $(X,\tau )$ is called adiscrete space.
Given $X=\mathbb {Z} ,$ the set of integers, the family $\tau$ of all finite subsets of the integers plus $\mathbb {Z}$ itself isnot a topology, because (for example) the union of all finite sets not containing zero is not finite and therefore not a member of the family of finite sets. The union of all finite sets not containing zero is also not all of $\mathbb {Z} ,$ and so it cannot be in $\tau .$

Definition via closed sets

Usingde Morgan's laws, the above axioms defining open sets become axioms definingclosed sets:

The empty set and $X {\displaystyle X}$ are closed.
The intersection of any collection of closed sets is also closed.
The union of any finite number of closed sets is also closed.

Using these axioms, another way to define a topological space is as a set $X {\displaystyle X}$ together with a collection $\tau$ of closed subsets of $X . {\displaystyle X.}$ Thus the sets in the topology $\tau$ are the closed sets, and their complements in $X {\displaystyle X}$ are the open sets.

Other definitions

There are many other equivalent ways to define a topological space: in other words the concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms.

Another way to define a topological space is by using theKuratowski closure axioms, which define the closed sets as thefixed points of anoperator on thepower set of $X . {\displaystyle X.}$

Anet is a generalisation of the concept ofsequence. A topology is completely determined if for every net in $X {\displaystyle X}$ the set of itsaccumulation points is specified.

Comparison of topologies

Main article:Comparison of topologies

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Many topologies can be defined on a set to form a topological space. When every open set of a topology $\tau _{1}$ is also open for a topology $\tau _{2},$ one says that $\tau _{2}$ isfiner than $\tau _{1},$ and $\tau _{1}$ iscoarser than $\tau _{2}.$ A proof that relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The termslarger andsmaller are sometimes used in place of finer and coarser, respectively. The termsstronger andweaker are also used in the literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading.

The collection of all topologies on a given fixed set $X {\displaystyle X}$ forms acomplete lattice: if $F=\left\{\tau _{\alpha }:\alpha \in A\right\}$ is a collection of topologies on $X, {\displaystyle X,}$ then themeet of $F {\displaystyle F}$ is the intersection of $F, {\displaystyle F,}$ and thejoin of $F {\displaystyle F}$ is the meet of the collection of all topologies on $X {\displaystyle X}$ that contain every member of $F . {\displaystyle F.}$

Continuous functions

Main article:Continuous function

Afunction $f:X\to Y$ between topological spaces is calledcontinuous if for every $x\in X$ and every neighbourhood $N {\displaystyle N}$ of $f(x)$ there is a neighbourhood $M {\displaystyle M}$ of $x {\displaystyle x}$ such that $f(M)\subseteq N.$ This relates easily to the usual definition in analysis. Equivalently, $f {\displaystyle f}$ is continuous if theinverse image of every open set is open.^[10] This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function. Ahomeomorphism is abijection that is continuous and whoseinverse is also continuous. Two spaces are calledhomeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.^[11]

Incategory theory, one of the fundamentalcategories isTop, which denotes thecategory of topological spaces whoseobjects are topological spaces and whosemorphisms are continuous functions. The attempt to classify the objects of this category (up to homeomorphism) byinvariants has motivated areas of research, such ashomotopy theory,homology theory, andK-theory.^{[citation needed]}

Examples of topological spaces

See also:List of topologies

A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space. Any set can be given thediscrete topology in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given thetrivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must beHausdorff spaces where limit points are unique.

There exist numerous topologies on any givenfinite set. Such spaces are calledfinite topological spaces. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.

Any set can be given thecofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallestT₁ topology on any infinite set.^[12]

Any set can be given thecocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.

The real line can also be given thelower limit topology. Here, the basic open sets are the half open intervals $[a,b).$ This topology on $\mathbb {R}$ is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.

If $\gamma$ is anordinal number, then the set $\gamma =[0,\gamma )$ may be endowed with theorder topology generated by the intervals $(\alpha ,\beta ),$ $[0,\beta ),$ and $(\alpha ,\gamma )$ where $\alpha$ and $\beta$ are elements of $\gamma .$

Everymanifold has anatural topology since it is locally Euclidean. Similarly, everysimplex and everysimplicial complex inherits a natural topology from .

TheSierpiński space is the simplest non-discrete topological space. It has important relations to thetheory of computation and semantics.

Topology from other topologies

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Every subset of a topological space can be given thesubspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For anyindexed family of topological spaces, the product can be given theproduct topology, which is generated by the inverse images of open sets of the factors under theprojection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space. This construction is a special case of aninitial topology.

Aquotient space is defined as follows: if $X {\displaystyle X}$ is a topological space and $Y {\displaystyle Y}$ is a set, and if $f:X\to Y$ is asurjective function, then the quotient topology on $Y {\displaystyle Y}$ is the collection of subsets of $Y {\displaystyle Y}$ that have openinverse images under $f . {\displaystyle f.}$ In other words, the quotient topology is the finest topology on $Y {\displaystyle Y}$ for which $f {\displaystyle f}$ is continuous. A common example of a quotient topology is when anequivalence relation is defined on the topological space $X . {\displaystyle X.}$ The map $f {\displaystyle f}$ is then the natural projection onto the set ofequivalence classes. This construction is a special case of afinal topology.

Metric spaces

Main article:Metric space

Metric spaces embody ametric, a precise notion of distance between points.

Everymetric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on anynormed vector space. On a finite-dimensionalvector space this topology is the same for all norms.

There are many ways of defining a topology on $\mathbb {R} ,$ the set ofreal numbers. The standard topology on $\mathbb {R}$ is generated by theopen intervals. The set of all open intervals forms abase or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, theEuclidean spaces $\mathbb {R} ^{n}$ can be given a topology. In theusual topology on $\mathbb {R} ^{n}$ the basic open sets are the openballs. Similarly, $\mathbb {C} ,$ the set ofcomplex numbers, and $\mathbb {C} ^{n}$ have a standard topology in which the basic open sets are open balls.

Topology from algebraic structure

For anyalgebraic objects we can introduce the discrete topology, under which the algebraic operations are continuous functions. For any such structure that is not finite, we often have a natural topology compatible with the algebraic operations, in the sense that the algebraic operations are still continuous. This leads to concepts such astopological groups,topological rings,topological fields andtopological vector spaces over the latter.Local fields are topological fields important innumber theory.

TheZariski topology is defined algebraically on thespectrum of a ring or analgebraic variety. On $\mathbb {R} ^{n}$ or $\mathbb {C} ^{n},$ the closed sets of the Zariski topology are thesolution sets of systems ofpolynomial equations.

Topological spaces with order structure

Spectral: A space isspectral if and only if it is the primespectrum of a ring (Hochster theorem).
Specialization preorder: In a space thespecialization preorder (orcanonical preorder) is defined by $x\leq y$ if and only if $\operatorname {cl} \{x\}\subseteq \operatorname {cl} \{y\},$ where $\operatorname {cl}$ denotes an operator satisfying theKuratowski closure axioms.

Topology from other structure

If $\Gamma$ is afilter on a set $X {\displaystyle X}$ then $\{\varnothing \}\cup \Gamma$ is a topology on $X . {\displaystyle X.}$

Many sets oflinear operators infunctional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.

Alinear graph has a natural topology that generalizes many of the geometric aspects ofgraphs withvertices andedges.

Outer space of afree group $F_{n}$ consists of the so-called "marked metric graph structures" of volume 1 on $F_{n}.$ ^[13]

Classification of topological spaces

Main article:Topological property

Topological spaces can be broadly classified,up to homeomorphism, by theirtopological properties. A topological property is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not homeomorphic it is sufficient to find a topological property not shared by them. Examples of such properties includeconnectedness,compactness, and variousseparation axioms. For algebraic invariants seealgebraic topology.

See also

Complete Heyting algebra – The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra.
Compact space – Type of mathematical space
Convergence space – Generalization of the notion of convergence that is found in general topology
Exterior space
Hausdorff space – Type of topological space
Hilbert space – Type of vector space in math
Hemicontinuity – Semicontinuity for set-valued functions
Linear subspace – In mathematics, vector subspace
Pointless topology
Quasitopological space – Function in topology
Relatively compact subspace – Subset of a topological space whose closure is compact
Space (mathematics) – Mathematical set with some added structure

Citations

^Schubert 1968, p. 13
^Sutherland, W. A. (1975).Introduction to metric and topological spaces. Oxford [England]: Clarendon Press.ISBN 0-19-853155-9.OCLC 1679102.
^^a ^bGallier & Xu 2013.
^J. Stillwell, Mathematics and its history
^"metric space".Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription orparticipating institution membership required.)
^Hausdorff, Felix (1914) [1914]. "Punktmengen in allgemeinen Räumen".Grundzüge der Mengenlehre. Göschens Lehrbücherei/Gruppe I: Reine und Angewandte Mathematik Serie (in German). Leipzig: Von Veit (published 2011). p. 211.ISBN 9783110989854. Retrieved20 August 2022.Unter einem m e t r i s c h e n R a u m e verstehen wir eine MengeE, [...].{{cite book}}:ISBN / Date incompatibility (help)
^Brown 2006, section 2.1.
^Brown 2006, section 2.2.
^Armstrong 1983, definition 2.1.
^Armstrong 1983, theorem 2.6.
^Munkres, James R (2015).Topology. Pearson. pp. 317–319.ISBN 978-93-325-4953-1.
^Anderson, B. A.; Stewart, D. G. (1969). " $T_{1}$ -complements of $T_{1}$ topologies".Proceedings of the American Mathematical Society.23:77–81.doi:10.2307/2037491.JSTOR 2037491.MR 0244927.
^Culler, Marc;Vogtmann, Karen (1986)."Moduli of graphs and automorphisms of free groups"(PDF).Inventiones Mathematicae.84 (1):91–119.Bibcode:1986InMat..84...91C.doi:10.1007/BF01388734.S2CID 122869546.

Bibliography

Armstrong, M. A. (1983) [1979].Basic Topology.Undergraduate Texts in Mathematics. Springer.ISBN 0-387-90839-0.
Bredon, Glen E.,Topology and Geometry (Graduate Texts in Mathematics), Springer; 1st edition (October 17, 1997).ISBN 0-387-97926-3.
Bourbaki, Nicolas;Elements of Mathematics: General Topology, Addison-Wesley (1966).
Brown, Ronald (2006).Topology and Groupoids. Booksurge.ISBN 1-4196-2722-8. (3rd edition of differently titled books)
Čech, Eduard;Point Sets, Academic Press (1969).
Fulton, William,Algebraic Topology, (Graduate Texts in Mathematics), Springer; 1st edition (September 5, 1997).ISBN 0-387-94327-7.
Gallier, Jean;Xu, Dianna (2013).A Guide to the Classification Theorem for Compact Surfaces. Springer.
Gauss, Carl Friedrich (1827).General investigations of curved surfaces.
Lipschutz, Seymour;Schaum's Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968).ISBN 0-07-037988-2.
Munkres, James;Topology, Prentice Hall; 2nd edition (December 28, 1999).ISBN 0-13-181629-2.
Runde, Volker;A Taste of Topology (Universitext), Springer; 1st edition (July 6, 2005).ISBN 0-387-25790-X.
Schubert, Horst (1968),Topology, Macdonald Technical & Scientific,ISBN 0-356-02077-0
Steen, Lynn A. andSeebach, J. Arthur Jr.;Counterexamples in Topology, Holt, Rinehart and Winston (1970).ISBN 0-03-079485-4.
Vaidyanathaswamy, R. (1999).Set Topology. Chelsea Publishing Co.ISBN 0486404560.
Willard, Stephen (2004).General Topology. Dover Publications.ISBN 0-486-43479-6.

External links

Wikiquote has quotations related toTopological space.

"Topological space",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

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