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
together with a collection
of subsets of
satisfying:
Given a topological space
one refers to the elements of
as the open sets of
and it is common only to refer to
in this way, or by the label topology. Then one makes the following secondary definitions:
- Given a second topological space a function is said to be continuous if and only if for every open subset of one has that is an open subset of
- A subset of is closed if and only if its complement is open.
- Given a subset of the closure is the set of all points such that any open set containing such a point must intersect
- Given a subset of the interior is the union of all open sets contained in
- Given an element of one says that a subset is a neighborhood of if and only if is contained in an open subset of which is also a subset of Some textbooks use “neighborhood of ” to instead refer to an open set containing
- One says that a net converges to a point of if for any open set containing the net is eventually contained in
- Given a set a filter is a collection of nonempty subsets of 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 defines a notion of a filter converging to a point of by requiring that any open set containing is an element of the filter.
- Given a set 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 one says that a filterbase converges to a point if every neighborhood of contains some element of the filterbase.
Definition via closed sets[edit]
Let
be a topological space. According to De Morgan’s laws, the collection
of closed sets satisfies the following properties:
Now suppose that
is only a set. Given any collection
of subsets of
which satisfy the above axioms, the corresponding set
is a topology on
and it is the only topology on
for which
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
the closure can be considered as a map
where
denotes the power set of
One has the following Kuratowski closure axioms:
If
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
all definitions can be phrased in terms of the closure operator:
Definition via interior operators[edit]
Given a topological space
the interior can be considered as a map
where
denotes the power set of
It satisfies the following conditions:
If
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
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
is a set and one declares a nonempty collection of neighborhoods for every point of
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
all definitions can be phrased in terms of neighborhoods:
Definition via convergence of nets[edit]
Convergence of nets satisfies the following properties:
- Every constant net converges to itself.
- Every subnet of a convergent net converges to the same limits.
- If a net does not converge to a point then there is a subnet such that no further subnet converges to Equivalently, if is a net such that every one of its subnets has a sub-subnet that converges to a point then converges to
- Diagonal principle/Convergence of iterated limits. If in and for every index is a net that converges to in then there exists a diagonal (sub)net of that converges to
- A diagonal net is refers to any subnet of
- The notation denotes the net defined by whose domain is the set ordered lexicographically first by and then by explicitly, given any two pairs declare that holds if and only if both (1) and also (2) if then
If
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
is defined by sending any given set
to the set of all limits of all nets valued in
the corresponding topology is the unique topology inducing the given convergences of nets to points.
Given a subset
of a topological space
A function
between two topological spaces is continuous if and only if for every
and every net
in
that converges to
in
the net
[note 1] converges to
in
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.
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