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]}

- The empty set and are in
- The union of any collection of sets in is also in
- The intersection of any pair of sets in is also in Equivalently, the intersection of any finite collection of sets in is also in

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^{[4]} - A subset of is
**closed**if and only if its complement is open.^{[5]} - Given a subset of the
**closure**is the set of all points such that any open set containing such a point must intersect^{[6]} - Given a subset of the
**interior**is the union of all open sets contained in^{[7]} - 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^{[8]}Some textbooks use "neighborhood of " to instead refer to an open set containing^{[9]} - One says that a
**net converges to a point**of if for any open set containing the net is eventually contained in^{[10]} - Given a set a filter is a collection of nonempty subsets of that is closed under finite intersection and under supersets.
^{[11]}Some textbooks allow a filter to contain the empty set, and reserve the name "proper filter" for the case in which it is excluded.^{[12]}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.^{[13]} - 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.
^{[14]}Given a topology on one says that a filterbase converges to a point if every neighborhood of contains some element of the filterbase.^{[15]}

Let be a topological space. According to De Morgan's laws, the collection of closed sets satisfies the following properties:^{[16]}

- The empty set and are elements of
- The intersection of any collection of sets in is also in
- The union of any pair of sets in is also in

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 second topological space a function is continuous if and only if for every closed subset of the set is closed as a subset of
^{[18]} - a subset of is open if and only if its complement is closed.
^{[19]} - given a subset of the closure is the intersection of all closed sets containing
^{[20]} - given a subset of the interior is the complement of the intersection of all closed sets containing

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 second topological space a function is continuous if and only if for every subset of one has that the set is a subset of
^{[23]} - A subset of is open if and only if
^{[24]} - A subset of is closed if and only if
^{[25]} - Given a subset of the interior is the complement of
^{[26]}

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 topological spaces and a function is continuous if and only if for every subset of one has that the set is a subset of
^{[29]} - A set is open if and only if it equals its interior.
^{[30]} - The closure of a set is the complement of the interior of its complement.
^{[31]}

See also: Neighbourhood system and Neighbourhood (mathematics) |

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 is a neighborhood of then is an element of
- The intersection of two neighborhoods of is a neighborhood of Equivalently, the intersection of finitely many neighborhoods of is a neighborhood of
- If contains a neighborhood of then is a neighborhood of
- If is a neighborhood of then there exists a neighborhood of such that is a neighborhood of each point of .

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.^{[32]} It follows that on a topological space all definitions can be phrased in terms of neighborhoods:

- Given another topological space a map is continuous if and only for every element of and every neighborhood of the preimage is a neighborhood of
^{[33]} - A subset of is open if and only if it is a neighborhood of each of its points.
- Given a subset of the interior is the collection of all elements of such that is a neighbourhood of .
- Given a subset of the closure is the collection of all elements of such that every neighborhood of intersects
^{[34]}

See also: Net (mathematics) and Fréchet–Urysohn space |

Convergence of nets satisfies the following properties:^{[35]}^{[36]}

- 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*refers to any subnet of - The notation denotes the net defined by whose domain is the set ordered lexicographically first by and then by
^{[36]}explicitly, given any two pairs declare that holds if and only if both (1) and also (2) if then

- A

If is a set, then given a notion of net convergence (telling what nets converge to what points^{[36]}) 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.^{[35]}

Given a subset of a topological space

- is open in if and only if every net converging to an element of is eventually contained in
- the closure of in is the set of all limits of all convergent nets valued in
^{[37]}^{[36]} - is closed in if and only if there does not exist a net in that converges to an element of the complement
^{[38]}A subset is closed in if and only if every limit point of every convergent net in necessarily belongs to^{[39]}

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 ^{[40]}

See also: Filters in topology |

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):

- Given a second topological space a function is continuous if and only if it preserves convergence of prefilters.
^{[41]} - A subset of is open if and only if every filter converging to an element of contains
^{[42]} - A subset of is closed if and only if there does not exist a prefilter on which converges to a point in the complement
^{[43]} - Given a subset of the closure consists of all points for which there is a prefilter on converging to
^{[44]} - A subset of is a neighborhood of if and only if it is an element of every filter converging to
^{[42]}