[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.
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.
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.
of closed sets satisfies the following properties:[16] Now suppose that
[17] 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: Given a topological space
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.
all definitions can be phrased in terms of the closure operator: Given a topological space
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.
all definitions can be phrased in terms of the interior operator, for instance: Given a topological space
is a set equipped with a mapping satisfying the above properties, then we can define the interior operator and vice versa.
satisfies the interior operator axioms, and hence defines a topology.
all definitions can be phrased in terms of the exterior operator, for instance: Given a topological space
is a set equipped with a mapping satisfying the above properties, then we can define closure operator and vice versa.
satisfies closure axioms, and hence boundary operation defines a topology.
all definitions can be phrased in terms of the boundary operator, for instance: The derived set of a subset
,[34] the derived set uniquely defines a topology.
all definitions can be phrased in terms of derived sets, for instance: Recall that this article follows the convention that a neighborhood is not necessarily open.
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.
all definitions can be phrased in terms of neighborhoods: Convergence of nets satisfies the following properties:[39][40]
is a set, then given a notion of net convergence (telling what nets converge to what points[40]) satisfying the above four axioms, a closure operator on
[44] 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): Notes