In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the
category of topological spaces, often denoted
Top, is the category whose objects are
topological spaceTopological 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
morphismIn mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory...
s are continuous maps. This is a category because the
compositionIn 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 spaceTopological 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 theoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
is known as
categorical topology.
N.B. Some authors use the name
Top for the category with
topological manifoldIn 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 categoryIn 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
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
s preserving this structure. There is a natural
forgetful functorIn 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 setsIn mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics.-Properties of the category of sets:...
which assigns to each topological space the underlying set and to each continuous map the underlying
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
.
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 inverseA 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 set of all topologies on a given set
X (called the
fiberIn mathematics, the fiber of a point y in Y under a function f : X → Y is the inverse image of {y} under f, that is, In a variant phrase, this is also called the fiber of f at y...
of
U above
X) forms a
complete latticeIn 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 elementIn 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. 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 topologyIn general topology and related areas of mathematics, the initial topology on a set , with respect to a family of functions on , 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 cocompleteIn 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...
, which means that all small
limits and colimitIn 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
diagramIn category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. The primary difference is that in the categorical setting one has morphisms as well: an indexed family of sets is a collection of sets, indexed by a fixed set , while a diagram is a...
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 topologyIn general topology and related areas of mathematics, the initial topology on a set , with respect to a family of functions on , 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 topologyIn general topology and related areas of mathematics, the final topology on a set , with respect to a family of functions into , is the finest topology on X which makes those functions continuous....
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
conesIn 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
In mathematics, and more specifically set theory, the empty set is the unique set having no members; its size 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 objectIn 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 singletonIn mathematics, a singleton is a set with exactly one element. For example, the set {0} is a singleton.-Properties:Note that a set such as is also a singleton: the only element is a set ....
topological space is a terminal object. There are thus no zero objects in Top.
- The product
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 topologyIn 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 productIn mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept....
. The coproduct is given by the disjoint unionIn 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
Equalizer or equaliser may refer to:*An equalization filter, an audio processing tool used for equalization*An Gain Equalizer, passive microwave components used to correct system slope problems in equalization...
of a pair of morphisms is given by placing the subspace topologyIn 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 of , the subspace...
on the set-theoretic equalizer. Dually, the coequalizerIn 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
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 we will understand objects to be...
s and inverse limitIn 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 topologyIn general topology and related areas of mathematics, the final topology on a set , with respect to a family of functions into , is the finest topology on X which makes those functions continuous....
and initial topologyIn general topology and related areas of mathematics, the initial topology on a set , with respect to a family of functions on , 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
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 pushoutsIn 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 .The pushout is the categorical dual of the...
in Top.
Other properties
- The 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 ....
s in Top are the injective continuous maps, the epimorphismIn 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 isomorphismIn abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings....
s are the homeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between two topological spaces that has a continuous inverse function...
s.
- The extremal monomorphisms are (up to isomorphism) the subspace
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 of , the subspace...
embeddings. Every extremal monomorphism is regular.
- The extremal epimorphisms are (essentially) the quotient maps. Every extremal epimorphism is regular.
- There are no zero morphism
In category theory, a zero morphism is a special kind of "trivial" morphism. Suppose C is a category, and for any two objects X and Y in C we are given a morphism 0
XY : X → Y with the following property: for any two morphism f : R → S and g : U → V we obtain a commutative...
s in Top, and in particular the category is not preadditiveIn 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
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 toposIn 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 objectIn 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
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 categoryIn 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
In topology and related branches of mathematics, a Hausdorff space, separated space or T
2 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...
property as a full subcategory. It should be noted that the added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with denseIn topology and related areas of mathematics, a subset A of a topological space X is called dense if any point in X can be "well-approximated" by points in A...
imagesIn mathematics, the image of a subset of a function's domain under the function is the set of all outputs obtained when the function is evaluated at each element of the subset...
in their codomainIn mathematics, the codomain, or target set, of a function is the set Y into which all of the output of the function is constrained to fall...
s, so that epimorphisms need not be surjective.