In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the category of topological spaces, often denoted Top, is the category whose objects are topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s and whose morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

s are continuous maps. This is a category because the composition

Function composition

In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

 of two continuous maps is again continuous. The study of Top and of properties of topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s using the techniques of category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

 is known as categorical topology.

N.B. Some authors use the name Top for the category with topological manifold

Topological manifold

In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...

s as objects and continuous maps as morphisms.

As a concrete category

Like many categories, the category Top is a concrete category

Concrete category

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...

 (also known as a construct), meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s preserving this structure. There is a natural forgetful functor

Forgetful functor

In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...

U : Top → Set

to the category of sets

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 all functions from A to B...

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

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

.

The forgetful functor U has both a left adjoint

D : Set → Top

which equips a given set with the discrete topology and a right adjoint

I : Set → Top

which equips a given set with the indiscrete topology. Both of these functors are, in fact, right inverse

Right inverse

A right inverse in mathematics may refer to:* A right inverse element with respect to a binary operation on a set* A right inverse function for a mapping between sets...

s to U (meaning that UD and UI are equal to the identity functor on Set). Moreover, since any function between discrete or indiscrete spaces is continuous, both of these functors give full embeddings of Set into Top.

The construct Top is also fiber-complete meaning that the category of all topologies on a given set X (called the fiber of U above X) forms a complete lattice

Complete lattice

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...

 when ordered by inclusion. The greatest element

Greatest element

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...

 in this fiber is the discrete topology on X while the least element is the indiscrete topology.

The construct Top is the model of what is called a topological category

Topological category

In mathematics, a topological category is a category that is enriched over the category of compactly generated Hausdorff spaces. They can be used as a foundation for higher category theory ....

. These categories are characterized by the fact that every structured source  has a unique initial lift . In Top the initial lift is obtained by placing the initial topology

Initial topology

In general topology and related areas of mathematics, the initial topology on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous.The subspace topology and product topology constructions are both special cases of initial...

 on the source. Topological categories have many nice 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

Complete category

In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C where J is small has a limit in C. Dually, a cocomplete category is one in which all small colimits exist...

, which means that all small limits and colimit

Limit (category theory)

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....

s 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

Diagram (category theory)

In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function...

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

Initial topology

In general topology and related areas of mathematics, the initial topology on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous.The subspace topology and product topology constructions are both special cases of initial...

 on (L, φ). Dually, colimits in Top are obtained by placing the final topology

Final topology

In general topology and related areas of mathematics, the final topology on a set X, with respect to a family of functions into X, is the finest topology on X which makes those functions continuous.- Definition :Given a set X and a family of topological spaces Y_i with functionsf_i: Y_i \to Xthe...

 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

Cone (category theory)

In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.-Definition:...

 in Top covering universal cones in Set.

Examples of limits and colimits in Top include:

• The empty set
  Empty set
  In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
  
   (considered as a topological space) is the initial object
  Initial object
  In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
  
   of Top; any singleton topological space is a terminal object. There are thus no zero objects in Top.
• The product
  Product (category theory)
  In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
  
   in Top is given by the product topology
  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...
  
   on the Cartesian product
  Cartesian product
  In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
  
  . The coproduct is given by the disjoint union
  Disjoint union (topology)
  In general topology and related areas of mathematics, the disjoint union of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology...
  
   of topological spaces.
• The equalizer of a pair of morphisms is given by placing the subspace topology
  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 natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
  
   on the set-theoretic equalizer. Dually, the coequalizer
  Coequalizer
  In category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category...
  
   is given by placing the quotient topology on the set-theoretic coequalizer.
• Direct limit
  Direct limit
  In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
  
  s and inverse limit
  Inverse limit
  In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
  
  s are the set-theoretic limits with the final topology
  Final topology
  In general topology and related areas of mathematics, the final topology on a set X, with respect to a family of functions into X, is the finest topology on X which makes those functions continuous.- Definition :Given a set X and a family of topological spaces Y_i with functionsf_i: Y_i \to Xthe...
  
   and initial topology
  Initial topology
  In general topology and related areas of mathematics, the initial topology on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous.The subspace topology and product topology constructions are both special cases of initial...
  
   respectively.
• Adjunction space
  Adjunction space
  In mathematics, an adjunction space is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be a topological spaces with A a subspace of Y. Let f : A → X be a continuous map...
  
  s are an example of pushouts
  Pushout (category theory)
  In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain: it is the colimit of the span X \leftarrow Z \rightarrow Y.The pushout is the...
  
   in Top.

Other properties

• The monomorphism
  Monomorphism
  In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
  
  s in Top are the injective continuous maps, the epimorphism
  Epimorphism
  In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
  
  s are the surjective continuous maps, and the isomorphism
  Isomorphism
  In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...
  
  s are the homeomorphism
  Homeomorphism
  In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
  
  s.
• The extremal monomorphisms are (up to isomorphism) the subspace
  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 natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
  
   embeddings. Every extremal monomorphism is regular.
• The extremal epimorphisms are (essentially) the quotient maps. Every extremal epimorphism is regular.
• There are no zero morphism
  Zero morphism
  In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...
  
  s in Top, and in particular the category is not preadditive
  Preadditive category
  In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
  
  .
• Top is not cartesian closed
  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 mathematical logic and the theory of programming, in...
  
   (and therefore also not a topos
  Topos
  In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
  
  ) since it does not have exponential object
  Exponential object
  In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories...
  
  s for all spaces.

Relationships to other categories

• The category of pointed topological spaces Top_• is a coslice category over Top.
• The homotopy category
  Homotopy category of topological spaces
  In mathematics, a homotopy category is a category whose objects are topological spaces and whose morphisms are homotopy classes of continuous functions. The homotopy category of all topological spaces is often denoted hTop or Toph....
  
   hTop has topological spaces for objects and homotopy equivalence classes of continuous maps for morphisms. This is a quotient category
  Quotient category
  In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. The notion is similar to that of a quotient group or quotient space, but in the categorical setting.-Definition:Let C be a category...
  
   of Top. One can likewise form the pointed homotopy category hTop_•.
• Top contains the important category Haus of topological spaces with the Hausdorff
  Hausdorff space
  In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
  
   property as a full subcategory. The added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with dense
  Dense set
  In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
  
   images
  Image (mathematics)
  In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
  
   in their codomain
  Codomain
  In mathematics, the codomain or target set 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...
  
  s, so that epimorphisms need not be surjective.