Axiomatic foundations of topological spaces

In the mathematical field of topology, a topological space is usually defined by declaring its open sets. However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a topological space determines a class of closed sets, of closure and interior operators, and of convergence of various types of objects. Each of these can instead be taken as the primary class of objects, with all of the others (including the class of open sets) directly determined from that new starting point. For example, in Kazimierz Kuratowski’s well-known textbook on point-set topology, a topological space is defined as a set together with a certain type of “closure operator,” and all other concepts are derived therefrom. Likewise, the neighborhood-based axioms (in the context of Hausdorff spaces) can be retraced to Felix Hausdorff’s original definition of a topological space in Grundzüge der Mengenlehre.^{[citation needed]}

Many different textbooks use many different inter-dependences of concepts to develop point-set topology. The result is always the same collection of objects: open sets, closed sets, and so on. For many practical purposes, the question of which foundation is chosen is irrelevant, as long as the meaning and interrelation between objects (many of which are given in this article), which are the same regardless of choice of development, are understood. However, there are cases where it can be useful to have flexibility. For instance, there are various natural notions of convergence of measures, and it is not immediately clear whether they arise from a topological structure or not. Such questions are greatly clarified by the topological axioms based on convergence.

Standard definitions via open sets[edit]

A topological space is a set

${displaystyle X}$

together with a collection

${displaystyle S}$

of subsets of

${displaystyle X}$

satisfying:

Given a topological space

${displaystyle (X,S),}$

one refers to the elements of

${displaystyle S}$

as the open sets of

${displaystyle X,}$

and it is common only to refer to

${displaystyle S}$

in this way, or by the label topology. Then one makes the following secondary definitions:

• Given a second topological space
  ${displaystyle Y,}$

  a function

  ${displaystyle f:Xto Y}$

  is said to be continuous if and only if for every open subset

  ${displaystyle U}$

  of

  ${displaystyle Y,}$

  one has that

  ${displaystyle f^{-1}(U)}$

  is an open subset of

  ${displaystyle X.}$

  

• A subset
  ${displaystyle C}$

  of

  ${displaystyle X}$

  is closed if and only if its complement

  ${displaystyle Xsetminus C}$

  is open.

• Given a subset
  ${displaystyle A}$

  of

  ${displaystyle X,}$

  the closure is the set of all points such that any open set containing such a point must intersect

  ${displaystyle A.}$

  

• Given a subset
  ${displaystyle A}$

  of

  ${displaystyle X,}$

  the interior is the union of all open sets contained in

  ${displaystyle A.}$

  

• Given an element
  ${displaystyle x}$

  of

  ${displaystyle X,}$

  one says that a subset

  ${displaystyle A}$

  is a neighborhood of

  ${displaystyle x}$

  if and only if

  ${displaystyle x}$

  is contained in an open subset of

  ${displaystyle X}$

  which is also a subset of

  ${displaystyle A.}$

  Some textbooks use “neighborhood of

  ${displaystyle x}$

  ” to instead refer to an open set containing

  ${displaystyle x.}$

  

• One says that a net converges to a point
  ${displaystyle x}$

  of

  ${displaystyle X}$

  if for any open set

  ${displaystyle U}$

  containing

  ${displaystyle x,}$

  the net is eventually contained in

  ${displaystyle U.}$

  

• Given a set
  ${displaystyle X,}$

  a filter is a collection of nonempty subsets of

  ${displaystyle X}$

  that is closed under finite intersection and under supersets. Some textbooks allow a filter to contain the empty set, and reserve the name “proper filter” for the case in which it is excluded. A topology on

  ${displaystyle X}$

  defines a notion of a filter converging to a point

  ${displaystyle x}$

  of

  ${displaystyle X,}$

  by requiring that any open set

  ${displaystyle U}$

  containing

  ${displaystyle x}$

  is an element of the filter.

• Given a set
  ${displaystyle X,}$

  a filterbase is a collection of nonempty subsets such that every two subsets intersect nontrivially and contain a third subset in the intersection. Given a topology on

  ${displaystyle X,}$

  one says that a filterbase converges to a point

  ${displaystyle x}$

  if every neighborhood of

  ${displaystyle x}$

  contains some element of the filterbase.

Definition via closed sets[edit]

Let

${displaystyle X}$

be a topological space. According to De Morgan’s laws, the collection

${displaystyle T}$

of closed sets satisfies the following properties:

Now suppose that

${displaystyle X}$

is only a set. Given any collection

${displaystyle T}$

of subsets of

${displaystyle X}$

which satisfy the above axioms, the corresponding set

${displaystyle {U:Xsetminus Uin T}}$

is a topology on

${displaystyle X,}$

and it is the only topology on

${displaystyle X}$

for which

${displaystyle T}$

is the corresponding collection of closed sets. This is to say that a topology can be defined by declaring the closed sets. As such, one can rephrase all definitions to be in terms of closed sets:

Definition via closure operators[edit]

Given a topological space

${displaystyle X,}$

the closure can be considered as a map

${displaystyle wp (X)to wp (X),}$

where

${displaystyle wp (X)}$

denotes the power set of

${displaystyle X.}$

One has the following Kuratowski closure axioms:

• 
  ${displaystyle Asubseteq operatorname {cl} (A)}$

  

• 
  ${displaystyle operatorname {cl} (operatorname {cl} (A))=operatorname {cl} (A)}$

  

• 
  ${displaystyle operatorname {cl} (Acup B)=operatorname {cl} (A)cup operatorname {cl} (B)}$

  

• 
  ${displaystyle operatorname {cl} (varnothing )=varnothing }$

  

If

${displaystyle X}$

is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of cl satisfies the previous axioms for closed sets, and hence defines a topology; it is the unique topology whose associated closure operator coincides with the given cl. As before, it follows that on a topological space

${displaystyle X,}$

all definitions can be phrased in terms of the closure operator:

Definition via interior operators[edit]

Given a topological space

${displaystyle X,}$

the interior can be considered as a map

${displaystyle wp (X)to wp (X),}$

where

${displaystyle wp (X)}$

denotes the power set of

${displaystyle X.}$

It satisfies the following conditions:

• 
  ${displaystyle operatorname {int} (A)subseteq A}$

  

• 
  ${displaystyle operatorname {int} (operatorname {int} (A))=operatorname {int} (A)}$

  

• 
  ${displaystyle operatorname {int} (Acap B)=operatorname {int} (A)cap operatorname {int} (B)}$

  

• 
  ${displaystyle operatorname {int} (X)=X}$

  

If

${displaystyle X}$

is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of int satisfies the previous axioms for open sets, and hence defines a topology; it is the unique topology whose associated interior operator coincides with the given int. It follows that on a topological space

${displaystyle X,}$

all definitions can be phrased in terms of the interior operator, for instance:

Definition via neighbourhoods[edit]

Recall that this article follows the convention that a neighborhood is not necessarily open. In a topological space, one has the following facts:

If

${displaystyle X}$

is a set and one declares a nonempty collection of neighborhoods for every point of

${displaystyle X,}$

satisfying the above conditions, then a topology is defined by declaring a set to be open if and only if it is a neighborhood of each of its points; it is the unique topology whose associated system of neighborhoods is as given. It follows that on a topological space

${displaystyle X,}$

all definitions can be phrased in terms of neighborhoods:

Definition via convergence of nets[edit]

Convergence of nets satisfies the following properties:

1. Every constant net converges to itself.
2. Every subnet of a convergent net converges to the same limits.
3. If a net does not converge to a point
   ${displaystyle x}$

   then there is a subnet such that no further subnet converges to

   ${displaystyle x.}$

   Equivalently, if

   ${displaystyle x_{bullet }}$

   is a net such that every one of its subnets has a sub-subnet that converges to a point

   ${displaystyle x,}$

   then

   ${displaystyle x_{bullet }}$

   converges to

   ${displaystyle x.}$

   

4. Diagonal principle/Convergence of iterated limits. If
   ${displaystyle left(x_{a}right)_{ain A}to x}$

   in

   ${displaystyle X}$

   and for every index

   ${displaystyle ain A,}$

   

   ${displaystyle left(x_{a}^{i}right)_{iin I_{a}}}$

   is a net that converges to

   ${displaystyle x_{a}}$

   in

   ${displaystyle X,}$

   then there exists a diagonal (sub)net of

   ${displaystyle left(x_{a}^{i}right)_{ain A,iin I_{a}}}$

   that converges to

   ${displaystyle x.}$

   

   • A diagonal net is refers to any subnet of
     ${displaystyle left(x_{a}^{i}right)_{ain A,iin I_{a}}.}$

     

   • The notation
     ${displaystyle left(x_{a}^{i}right)_{ain A,iin I_{a}}}$

     denotes the net defined by

     ${displaystyle (a,i)mapsto x_{a}^{i}}$

     whose domain is the set

     ${displaystyle {textstyle bigcup limits _{ain A}}Atimes I_{a}}$

     ordered lexicographically first by

     ${displaystyle A}$

     and then by

     ${displaystyle I_{a};}$

     explicitly, given any two pairs

     ${displaystyle (a,i),left(a_{2},i_{2}right)in {textstyle bigcup limits _{ain A}}Atimes I_{a},}$

     declare that

     ${displaystyle (a,i)leq left(a_{2},i_{2}right)}$

     holds if and only if both (1)

     ${displaystyle aleq a_{2},}$

     and also (2) if

     ${displaystyle a=a_{2}}$

     then

     ${displaystyle ileq i_{2}.}$

     

If

${displaystyle X}$

is a set, then given a notion of net convergence (telling what nets converge to what points) satisfying the above four axioms, a closure operator on

${displaystyle X}$

is defined by sending any given set

${displaystyle A}$

to the set of all limits of all nets valued in

${displaystyle A;}$

the corresponding topology is the unique topology inducing the given convergences of nets to points.

Given a subset

${displaystyle Asubseteq X}$

of a topological space

${displaystyle X:}$

A function

${displaystyle f:Xto Y}$

between two topological spaces is continuous if and only if for every

${displaystyle xin X}$

and every net

${displaystyle x_{bullet }}$

in

${displaystyle X}$

that converges to

${displaystyle x}$

in

${displaystyle X,}$

the net

${displaystyle fleft(x_{bullet }right)}$

^{[note 1]} converges to

${displaystyle f(x)}$

in

${displaystyle Y.}$

Definition via convergence of filters[edit]

A topology can also be defined on a set by declaring which filters converge to which points.^{[citation needed]} One has the following characterizations of standard objects in terms of filters and prefilters (also known as filterbases):

See also[edit]

Citations[edit]

Notes

References[edit]

• Dugundji, James (1978). Topology. Allyn and Bacon Series in Advanced Mathematics (Reprinting of the 1966 original ed.). Boston, Mass.-London-Sydney: Allyn and Bacon, Inc.
• Engelking, Ryszard (1977). General topology. Monografie Matematyczne. Vol. 60 (Translated by author from Polish ed.). Warsaw: PWN—Polish Scientific Publishers.
• Kelley, John L. (1975). General topology. Graduate Texts in Mathematics. Vol. 27 (Reprint of the 1955 ed.). New York-Berlin: Springer-Verlag.
• Kuratowski, K. (1966). Topology. Vol. I. (Translated from the French by J. Jaworowski. Revised and augmented ed.). New York-London/Warsaw: Academic Press/Państwowe Wydawnictwo Naukowe.
• Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.