In mathematics, the category of topological spaces, often denoted Top, is the

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

whose

object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...

s are

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 po ...

s and whose

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...

s are

continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...

s. This is a category because the

composition Composition or Compositions may refer to: Arts and literature * Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of

s using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with

topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout ma ...

s, with compactly generated spaces as objects and continuous maps as morphisms or with the

category of compactly generated weak Hausdorff spaces In mathematics, the category of compactly generated weak Hausdorff spaces CGWH is one of typically used categories in algebraic topology as a substitute for the category of topological spaces, as the latter lacks some of the pleasant properties one ...

As a concrete category

Like many categories, the category Top is a

concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects o ...

, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

s preserving this structure. There is a natural

forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sig ...

:''U'' : Top → Set to the

category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...

which assigns to each topological space the underlying set and to each continuous map the underlying

. The forgetful functor ''U'' has both a

left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

:''D'' : Set → Top which equips a given set with the

discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest t ...

, and a

right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...

:''I'' : Set → Top which equips a given set with the

indiscrete 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 conseque ...

. Both of these functors are, in fact, right inverses to ''U'' (meaning that ''UD'' and ''UI'' are equal to the

identity functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

on Set). Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give

full embedding In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...

s of Set into Top. Top is also ''fiber-complete'' meaning that the category of all topologies on a given set ''X'' (called the ''

fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...

'' of ''U'' above ''X'') forms a

complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' S ...

when ordered by inclusion. The

greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an el ...

in this fiber is the discrete topology on ''X'', while the

least element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an ele ...

is the indiscrete topology. Top is the model of what is called a

topological category In category theory, a discipline in mathematics, the notion of topological category has a number of different, inequivalent definitions. In one approach, a topological category is a category that is enriched over the category of compactly gene ...

. These categories are characterized by the fact that every

structured source Structuring, also known as smurfing in banking jargon, is the practice of executing financial transactions such as making bank deposits in a specific pattern, calculated to avoid triggering financial institutions to file reports required by law ...

(X \to UA_i)_I

has a unique

initial lift In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph tha ...

( A \to A_i)_I

. In Top the initial lift is obtained by placing the

initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' ...

on the source. Topological categories have many properties in common with Top (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).

Limits and colimits

The category Top is both complete and cocomplete, which means that all small limits and colimits exist in Top. In fact, the forgetful functor ''U'' : Top → Set uniquely lifts both limits and colimits and preserves them as well. Therefore, (co)limits in Top are given by placing topologies on the corresponding (co)limits in Set. Specifically, if ''F'' is a

diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three ...

in Top and (''L'', ''φ'' : ''L'' → ''F'') is a limit of ''UF'' in Set, the corresponding limit of ''F'' in Top is obtained by placing the

on (''L'', ''φ'' : ''L'' → ''F''). Dually, colimits in Top are obtained by placing the

final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that ...

on the corresponding colimits in Set. Unlike many ''algebraic'' categories, the forgetful functor ''U'' : Top → Set does not create or reflect limits since there will typically be non-universal

cones A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines co ...

in Top covering universal cones in Set. Examples of limits and colimits in Top include: *The

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...

(considered as a topological space) is the

initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...

of Top; any

singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance of ...

topological space is a

terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...

. There are thus no

zero object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

s in Top. *The

product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Prod ...

in Top is given by the

product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...

on the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...

. The

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...

is given by the

disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ...

of topological spaces. *The equalizer of a pair of morphisms is given by placing the

subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...

on the set-theoretic equalizer. Dually, the

coequalizer In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. Definition A coequalizer is a co ...

is given by placing the

quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...

on the set-theoretic coequalizer. *

Direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cat ...

s and

inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ...

s are the set-theoretic limits with the

and

respectively. *

Adjunction space In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let ''X'' and ''Y'' be topological spaces, and let ''A'' be a subspace of ' ...

s are an example of pushouts in Top.

Other properties

*The

monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphis ...

s in Top are the

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 contraposi ...

continuous maps, the

epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...

s are the

surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...

continuous maps, and the

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

s are the

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 isomor ...

s. *The extremal monomorphisms are (up to isomorphism) the subspace embeddings. In fact, in Top all extremal monomorphisms happen to satisfy the stronger property of being

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

. *The extremal epimorphisms are (essentially) the

quotient map In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient ...

s. Every extremal epimorphism is regular. *The split monomorphisms are (essentially) the inclusions of

retract Retraction or retract(ed) may refer to: Academia * Retraction in academic publishing, withdrawals of previously published academic journal articles Mathematics * Retraction (category theory) * Retract (group theory) * Retraction (topology) H ...

s into their ambient space. *The split epimorphisms are (up to isomorphism) the continuous surjective maps of a space onto one of its retracts. *There are no

zero morphism In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object. Definitions Suppose C is a category, and ''f'' : ''X'' → ''Y'' is a morphism in C. T ...

s in Top, and in particular the category is not preadditive. *Top is not

cartesian closed In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in math ...

(and therefore also not a

topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...

) since it does not have

exponential object In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed ca ...

s for all spaces. When this feature is desired, one often restricts to the full subcategory of

compactly generated Hausdorff space In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subsp ...

s CGHaus or the

. However, Top is contained in the exponential category of pseudotopologies, which is itself a subcategory of the (also exponential) category of

convergence space In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a that satisfies certain properties relating elements of ''X'' with the family of filters on ''X''. Convergence spaces generaliz ...

Relationships to other categories

*The category of

pointed topological space In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...

s Top_• is a

coslice category In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track ...

over Top. * The

homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed b ...

hTop has topological spaces for objects and homotopy equivalence classes of continuous maps for morphisms. This is a

quotient category In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but ...

of Top. One can likewise form the pointed homotopy category hTop_•. *Top contains the important category Haus of

Hausdorff spaces In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhood (mathematics), neighbourhoods of each which are disjoint s ...

as a

full subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...

. The added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with

dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

images An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...

in their

codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...

s, so that epimorphisms need not be

. *Top contains the full subcategory CGHaus of

s, which has the important property of being a

Cartesian closed category In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in ...

while still containing all of the typical spaces of interest. This makes CGHaus a particularly ''convenient category of topological spaces'' that is often used in place of Top. * The forgetful functor to Set has both a left and a right adjoint, as described above in the concrete category section. * There is a functor to the category of locales Loc sending a topological space to its locale of open sets. This functor has a right adjoint that sends each locale to its topological space of points. This adjunction restricts to an equivalence between the category of

sober space In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible closed subset of ''X'' is the closure of exactly one point of ''X'': that is, every irreducible closed subset has a unique generic point. Definiti ...

s and spatial locales. *The

homotopy hypothesis In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states that the ∞-groupoids are spaces. If we model our ∞-groupoids as Kan complexes, then the homotopy types of the geometric realizations of these sets give mod ...

relates Top with ∞Grpd, the category of ∞-groupoids. The conjecture states that ∞-groupoids are equivalent to topological spaces modulo

weak homotopy equivalence In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category. A model category is a category with cl ...

Citations

References

* Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990)
''Abstract and Concrete Categories''
(4.2MB PDF). Originally publ. John Wiley & Sons. . (now free on-line edition). * * * * Herrlich, Horst:
Topologische Reflexionen und Coreflexionen
'. Springer Lecture Notes in Mathematics 78 (1968). * Herrlich, Horst: ''Categorical topology 1971–1981''. In: General Topology and its Relations to Modern Analysis and Algebra 5, Heldermann Verlag 1983, pp. 279–383. * Herrlich, Horst & Strecker, George E.
Categorical Topology – its origins, as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971
In: Handbook of the History of General Topology (eds. C.E.Aull & R. Lowen), Kluwer Acad. Publ. vol 1 (1997) pp. 255–341. {{refend

Topological spaces 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 poi ...

General topology