Characterizations of the category of topological spaces
Encyclopedia
In mathematics
, a topological space
is usually defined in terms of open set
s. However, there are many equivalent characterization
s of the category of topological spaces
. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation.
, and every pair of these categories can be shown to be concretely isomorphic. This means that for every pair of categories defined below, there is an isomorphism of categories
, for which corresponding objects have the same underlying set
and corresponding morphism
s are identical as set functions.
To actually establish the concrete isomorphisms is more tedious than illuminating. The simplest approach is probably to construct pairs of inverse concrete isomorphisms between each category and the category of topological spaces
Top. This would involve the following:
s, i.e., all pairs (X,T) of set X together with a collection T of subset
s of X satisfying:
Morphisms: all ordinary continuous functions, i.e. all functions such that the inverse image of every open set is open.
Comments: This is the ordinary category of topological spaces
.
s of X satisfying:
Morphisms: all functions such that the inverse image of every closed set is closed.
Comments: This is the category that results by replacing each lattice
of open sets in a topological space by its order-theoretic dual
of closed sets, the lattice of complements of open sets. The relation between the two definitions is given by De Morgan's laws.
cl : P(X) → P(X) satisfying the Kuratowski closure axioms
:
Morphisms: all closure-preserving functions, i.e., all functions f between two closure spaces
Comments: The Kuratowski closure axioms abstract the properties of the closure operator on a topological space, which assigns to each subset its topological closure. This topological closure operator
has been generalized in category theory
; see Categorical Closure Operators by G. Castellini in "Categorical Perspectives", referenced below.
of the Kuratowski closure axioms
:
Morphisms: all interior-preserving functions, i.e., all functions f between two interior spaces
Comments: The interior operator assigns to each subset its topological interior, in the same way the closure operator assigns to each subset its topological closure.
s on X, satisfying for every x in X:
Morphisms: all neighbourhood-preserving functions, i.e., all functions f : (X, N) → (Y, N) such that if V is in N(f(x)), then there exists U in N(x) such that f(U) is contained in V. This is equivalent to asking that whenever V is in N(f(x)), then f−1(V) is in N(x).
Comments: This definition axiomatizes the notion of neighbourhood. We say that U is a neighbourhood of x if U is in N(x). The open sets can be recovered by declaring a set to be open if it is a neighbourhood of each of its points; the final axiom then states that every neighbourhood contains an open set. These axioms (coupled with the Hausdorff condition
) can be retraced to Felix Hausdorff
's original definition of a topological space in Grundzüge der Mengenlehre
.
relation between filters
on X and points of x. This definition demonstrates that convergence of filters can be viewed as a fundamental topological notion. A topology in the usual sense can be recovered by declaring a set A to be closed if, whenever F is a filter on A, then A contains all points to which F converges.
Similarly, Top can be described via net convergence. As for filters, this definition shows that convergence of nets can be viewed as a fundamental topological notion. A topology in the usual sense can be recovered by declaring a set A to be closed if, whenever (xα) is a net on A, then A contains all points to which (xα) converges.
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...
, a 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...
is usually defined in terms of open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s. However, there are many equivalent characterization
Characterization (mathematics)
In mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in...
s of the category of topological spaces
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 continuous...
. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation.
Definitions
Formally, each of the following definitions defines a concrete categoryConcrete 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...
, and every pair of these categories can be shown to be concretely isomorphic. This means that for every pair of categories defined below, there is an isomorphism of categories
Isomorphism of categories
In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D and GF = 1C. This means that both the objects and the morphisms of C and D stand in a one to one correspondence to each other...
, for which corresponding objects have the same underlying set
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...
and corresponding 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 identical as set functions.
To actually establish the concrete isomorphisms is more tedious than illuminating. The simplest approach is probably to construct pairs of inverse concrete isomorphisms between each category and the category of topological spaces
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 continuous...
Top. This would involve the following:
- Defining inverse object functions, checking that they are inverse, and checking that corresponding objects have the same underlying set.
- Checking that a set function is "continuous" (i.e., a morphism) in the given category if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is continuous (a morphism) in Top.
Definition via open sets
Objects: all topological spaceTopological 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, i.e., all pairs (X,T) of set X together with a collection T of subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of X satisfying:
- The empty setEmpty setIn 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...
and X are in T. - The unionUnion (set theory)In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of any collection of sets in T is also in T. - The intersectionIntersection (set theory)In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of any pair of sets in T is also in T.
- The sets in T are the open setOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s.
Morphisms: all ordinary continuous functions, i.e. all functions such that the inverse image of every open set is open.
Comments: This is the ordinary category of topological spaces
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 continuous...
.
Definition via closed sets
Objects: all pairs (X,T) of set X together with a collection T of subsetSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of X satisfying:
- The empty setEmpty setIn 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...
and X are in T. - The intersectionIntersection (set theory)In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of any collection of sets in T is also in T. - The unionUnion (set theory)In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of any pair of sets in T is also in T.
- The sets in T are the closed setClosed setIn geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
s.
Morphisms: all functions such that the inverse image of every closed set is closed.
Comments: This is the category that results by replacing each lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
of open sets in a topological space by its order-theoretic dual
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
of closed sets, the lattice of complements of open sets. The relation between the two definitions is given by De Morgan's laws.
Definition via closure operators
Objects: all pairs (X,cl) of set X together with a closure operatorClosure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
cl : P(X) → P(X) satisfying the Kuratowski closure axioms
Kuratowski closure axioms
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition...
:
-
(Extensivity) -
(Idempotence) -
(Preservation of binary unions) -
(Preservation of nullary unions)
Morphisms: all closure-preserving functions, i.e., all functions f between two closure spaces
- such that for all subsets
of 
Comments: The Kuratowski closure axioms abstract the properties of the closure operator on a topological space, which assigns to each subset its topological closure. This topological closure operator
Closure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
has been generalized in 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...
; see Categorical Closure Operators by G. Castellini in "Categorical Perspectives", referenced below.
Definition via interior operators
Objects: all pairs (X,int) of set X together with an interior operator int : P(X) → P(X) satisfying the following dualisationDuality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
of the Kuratowski closure axioms
Kuratowski closure axioms
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition...
:
-
-
(Idempotence) -
(Preservation of binary intersections) -
(Preservation of nullary intersections)
Morphisms: all interior-preserving functions, i.e., all functions f between two interior spaces
- such that for all subsets
of 
Comments: The interior operator assigns to each subset its topological interior, in the same way the closure operator assigns to each subset its topological closure.
Definition via neighbourhoods
Objects: all pairs (X,N) of set X together with a neighbourhood function N : X → F(X), where F(X) denotes the set of all filterFilter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...
s on X, satisfying for every x in X:
- If U is in N(x), then x is in U.
- If U is in N(x), then there exists V in N(x) such that U is in N(y) for all y in V.
Morphisms: all neighbourhood-preserving functions, i.e., all functions f : (X, N) → (Y, N) such that if V is in N(f(x)), then there exists U in N(x) such that f(U) is contained in V. This is equivalent to asking that whenever V is in N(f(x)), then f−1(V) is in N(x).
Comments: This definition axiomatizes the notion of neighbourhood. We say that U is a neighbourhood of x if U is in N(x). The open sets can be recovered by declaring a set to be open if it is a neighbourhood of each of its points; the final axiom then states that every neighbourhood contains an open set. These axioms (coupled with the Hausdorff condition
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...
) can be retraced to Felix Hausdorff
Felix Hausdorff
Felix Hausdorff was a Jewish German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.-Life:Hausdorff studied at the University of Leipzig,...
's original definition of a topological space in Grundzüge der Mengenlehre
Grundzüge der Mengenlehre
Grundzüge der Mengenlehre is an influential book on set theory written by Felix Hausdorff.First published in April 1914, Grundzüge der Mengenlehre was the first comprehensive introduction to set theory...
.
Definition via convergence
The category of topological spaces can also be defined via a convergenceConvergence
-Mathematics:* Convergence , refers to the notion that some functions and sequences approach a limit under certain conditions* Convergence , the notion that a sequence of transformations come to the same conclusion, no matter what order they are performed in.-Natural sciences:*Convergence ,...
relation between filters
Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...
on X and points of x. This definition demonstrates that convergence of filters can be viewed as a fundamental topological notion. A topology in the usual sense can be recovered by declaring a set A to be closed if, whenever F is a filter on A, then A contains all points to which F converges.
Similarly, Top can be described via net convergence. As for filters, this definition shows that convergence of nets can be viewed as a fundamental topological notion. A topology in the usual sense can be recovered by declaring a set A to be closed if, whenever (xα) is a net on A, then A contains all points to which (xα) converges.