NationMaster - Encyclopedia: Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first introduced by Kazimierz Kuratowski, in a slightly different form that applied only to Hausdorff spaces. A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This article is about a logical statement. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... Kazimierz Kuratowski (born February 2, 1896, Warsaw, died June 18, 1980, Warsaw) was a Polish mathematician. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator. In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

1 Definition
2 Notes
3 Recovering topological definitions
4 See also

Definition

A topological space $(X,operatorname{cl})$ is a set $X$ with a function

$operatorname{cl}:mathcal{P}(X) to mathcal{P}(X)$

called the closure operator where $mathcal{P}(X)$ is the power set of $X$ . In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...

The closure operator has to satisfy the following properties

$A subseteq operatorname{cl}(A) !$ (Extensivity)
$operatorname{cl}(operatorname{cl}(A)) = operatorname{cl}(A) !$ (Idempotence)
$operatorname{cl}(A cup B) = operatorname{cl}(A) cup operatorname{cl}(B) !$ (Preservation of binary unions)
$operatorname{cl}(varnothing) = varnothing !$ (Preservation of nullary unions)

If the second axiom, that of idempotence, is relaxed, then the axioms define a praclosure. In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In topology, a praclosure operator, or ÄŒech closure operator is a map between subsets of a set, similar to a closure operator, except that it is not required to be idempotent. ...

Notes

Axioms (3) and (4) can be generalised (using a proof by mathematical induction) to the equivalent single statement: Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

$operatorname{cl}(A_{1} cup cdots cup A_{n}) = operatorname{cl}(A_{1}) cup cdots cup operatorname{cl}(A_{n}), n geq 0 !$ (Preservation of finitary unions).

Recovering topological definitions

A function between two topological spaces

$f:(X,operatorname{cl}) to (X',operatorname{cl}')$

is called continuous if for all subsets $A$ of $X$ In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

$f(operatorname{cl}(A)) subset operatorname{cl}(f(A))$

A point $p$ is called close to $A$ in $(X,operatorname{cl})$ if $pin operatorname{cl}(A)$ In topology and related areas in mathematics closeness is one of the basic concepts in a topological space. ...

$A$ is called closed in $(X,operatorname{cl})$ if $A=operatorname{cl}(A)$ . In other words the closed sets of $X$ are the fixed points of the closure operator. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...

Closure (mathematics) Summary (1481 words)
	An operation of a different sort is that of finding the limit points of a subset of a topological space (if the space is first countable, it suffices to restrict consideration to the limits of sequences but in general one must consider at least limits of nets).
	In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X is closed under the operation of linear combination.
	In group theory, the normal closure of a set of group elements is the smallest normal subgroup containing the set.

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Contents

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Notes

Recovering topological definitions

See also

Definition