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.

Standard definitions via open sets

A topological space is a set ${\displaystyle X}$ together with a collection ${\displaystyle S}$ of subsets of ${\displaystyle X}$ satisfying:^[3]

• The empty set and ${\displaystyle X}$ are in ${\displaystyle S.}$
• The union of any collection of sets in ${\displaystyle S}$ is also in ${\displaystyle S.}$
• The intersection of any pair of sets in ${\displaystyle S}$ is also in ${\displaystyle S.}$ Equivalently, the intersection of any finite collection of sets in ${\displaystyle S}$ is also in ${\displaystyle S.}$

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

• Given a second topological space ${\displaystyle Y,}$ a function ${\displaystyle f:X\to Y}$ is said to be continuous if and only if for every open subset ${\displaystyle U}$ of ${\displaystyle Y,}$ one has that ${\displaystyle f^{-1}(U)}$ is an open subset of ${\displaystyle X.}$^[4]
• A subset ${\displaystyle C}$ of ${\displaystyle X}$ is closed if and only if its complement ${\displaystyle X\setminus C}$ is open.^[5]
• Given a subset ${\displaystyle A}$ of ${\displaystyle X,}$ the closure is the set of all points such that any open set containing such a point must intersect ${\displaystyle A.}$^[6]
• Given a subset ${\displaystyle A}$ of ${\displaystyle X,}$ the interior is the union of all open sets contained in ${\displaystyle A.}$^[7]
• Given an element ${\displaystyle x}$ of ${\displaystyle X,}$ one says that a subset ${\displaystyle A}$ is a neighborhood of ${\displaystyle x}$ if and only if ${\displaystyle x}$ is contained in an open subset of ${\displaystyle X}$ which is also a subset of ${\displaystyle A.}$^[8] Some textbooks use "neighborhood of ${\displaystyle x}$" to instead refer to an open set containing ${\displaystyle x.}$^[9]
• One says that a net converges to a point ${\displaystyle x}$ of ${\displaystyle X}$ if for any open set ${\displaystyle U}$ containing ${\displaystyle x,}$ the net is eventually contained in ${\displaystyle U.}$^[10]
• Given a set ${\displaystyle X,}$ a filter is a collection of nonempty subsets of ${\displaystyle X}$ 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 ${\displaystyle X}$ defines a notion of a filter converging to a point ${\displaystyle x}$ of ${\displaystyle X,}$ by requiring that any open set ${\displaystyle U}$ containing ${\displaystyle x}$ is an element of the filter.^[13]
• Given a set ${\displaystyle X,}$ 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 ${\displaystyle X,}$ one says that a filterbase converges to a point ${\displaystyle x}$ if every neighborhood of ${\displaystyle x}$ contains some element of the filterbase.^[15]

Definition via closed sets

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

• The empty set and ${\displaystyle X}$ are elements of ${\displaystyle T}$
• The intersection of any collection of sets in ${\displaystyle T}$ is also in ${\displaystyle T.}$
• The union of any pair of sets in ${\displaystyle T}$ is also in ${\displaystyle T.}$

Now suppose that ${\displaystyle X}$ is only a set. Given any collection ${\displaystyle T}$ of subsets of ${\displaystyle X}$ which satisfy the above axioms, the corresponding set ${\displaystyle \{U:X\setminus U\in T\}}$ is a topology on ${\displaystyle X,}$ and it is the only topology on ${\displaystyle X}$ for which ${\displaystyle T}$ 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 ${\displaystyle Y,}$ a function ${\displaystyle f:X\to Y}$ is continuous if and only if for every closed subset ${\displaystyle U}$ of $$Zdroj:https://en.wikipedia.org?pojem=Characterizations_of_the_category_of_topological_spaces
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