Axiomatic foundations of topological spaces

In the mathematical field of topology, a topological space is usually defined by declaring its open sets.[1] 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.[2] 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.

A topological space is a set X together with a collection S of subsets of X satisfying:[3]

Given a topological space (X, S), one refers to the elements of S as the open sets of X, and it is common to only refer to S in this way, or by the label topology. Then one makes the following secondary definitions:

Let X be a topological space. According to De Morgan's laws, the collection T of closed sets satisfies the following properties:[16]

Given a topological space X, the closure can be considered as a map P(X) → P(X), where P(X) denotes the power set of X. One has the following Kuratowski closure axioms:[21]

If 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.[22] As before, it follows that on a topological space X, all definitions can be phrased in terms of the closure operator:

Given a topological space X, the interior can be considered as a map P(X) → P(X), where P(X) denotes the power set of X. It satisfies the following conditions:[27]

If 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.[28] It follows that on a topological space X, all definitions can be phrased in terms of the interior operator, for instance:

Recall that this article follows the convention that a neighborhood is not necessarily open. In a topological space, one has the following facts:[32]

If X is a set and one declares a nonempty collection of neighborhoods for every point of 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.[33] It follows that on a topological space X, all definitions can be phrased in terms of neighborhoods:

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 filterbases: