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 together with a collection of subsets of satisfying: [3]
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:
Let be a topological space. According to De Morgan's laws, the collection of closed sets satisfies the following properties: [16]
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. [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 the closure can be considered as a map where denotes the power set of One has the following Kuratowski closure axioms: [21]
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. [22] As before, it follows that on a topological space all definitions can be phrased in terms of the closure operator:
Given a topological space the interior can be considered as a map where denotes the power set of It satisfies the following conditions: [27]
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. [28] It follows that on a topological space all definitions can be phrased in terms of the interior operator, for instance:
Given a topological space the boundary can be considered as a map where denotes the power set of It satisfies the following conditions: [32]
If is a set equipped with a mapping satisfying the above properties, then we can define the interior operator and vice versa. More precisely, if we define , satisfies the interior operator axioms, and hence defines a topology. [33] Conversely, if we define , satisfies the above axioms. Moreover, these correspondence is 1-1. It follows that on a topological space all definitions can be phrased in terms of the exterior operator, for instance:
Given a topological space the boundary can be considered as a map where denotes the power set of It satisfies the following conditions: [32]
If is a set equipped with a mapping satisfying the above properties, then we can define closure operator and vice versa. More precisely, if we define , satisfies closure axioms, and hence boundary operation defines a topology. Conversely, if we define , satisfies above axioms. Moreover, these correspondence is 1-1. It follows that on a topological space all definitions can be phrased in terms of the boundary operator, for instance:
The derived set of a subset of a topological space is the set of all points that are limit points of that is, points such that every neighbourhood of contains a point of other than itself. The derived set of , denoted , satisfies the following conditions: [32]
Since a set is closed if and only if , [34] the derived set uniquely defines a topology. It follows that on a topological space 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. In a topological space, one has the following facts: [36]
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. [36] It follows that on a topological space all definitions can be phrased in terms of neighborhoods:
Convergence of nets satisfies the following properties: [39] [40]
If 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 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. [39]
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 [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
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 together with a collection of subsets of satisfying: [3]
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:
Let be a topological space. According to De Morgan's laws, the collection of closed sets satisfies the following properties: [16]
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. [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 the closure can be considered as a map where denotes the power set of One has the following Kuratowski closure axioms: [21]
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. [22] As before, it follows that on a topological space all definitions can be phrased in terms of the closure operator:
Given a topological space the interior can be considered as a map where denotes the power set of It satisfies the following conditions: [27]
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. [28] It follows that on a topological space all definitions can be phrased in terms of the interior operator, for instance:
Given a topological space the boundary can be considered as a map where denotes the power set of It satisfies the following conditions: [32]
If is a set equipped with a mapping satisfying the above properties, then we can define the interior operator and vice versa. More precisely, if we define , satisfies the interior operator axioms, and hence defines a topology. [33] Conversely, if we define , satisfies the above axioms. Moreover, these correspondence is 1-1. It follows that on a topological space all definitions can be phrased in terms of the exterior operator, for instance:
Given a topological space the boundary can be considered as a map where denotes the power set of It satisfies the following conditions: [32]
If is a set equipped with a mapping satisfying the above properties, then we can define closure operator and vice versa. More precisely, if we define , satisfies closure axioms, and hence boundary operation defines a topology. Conversely, if we define , satisfies above axioms. Moreover, these correspondence is 1-1. It follows that on a topological space all definitions can be phrased in terms of the boundary operator, for instance:
The derived set of a subset of a topological space is the set of all points that are limit points of that is, points such that every neighbourhood of contains a point of other than itself. The derived set of , denoted , satisfies the following conditions: [32]
Since a set is closed if and only if , [34] the derived set uniquely defines a topology. It follows that on a topological space 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. In a topological space, one has the following facts: [36]
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. [36] It follows that on a topological space all definitions can be phrased in terms of neighborhoods:
Convergence of nets satisfies the following properties: [39] [40]
If 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 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. [39]
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 [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