Kuratowski closure axioms

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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related branches of mathematics, the Kuratowski closure axioms are a set of
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s that can be used to define a
topological structure In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
on a set. They are equivalent to the more commonly used
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
definition. They were first formalized by
Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Biography and studies Kazimierz Kuratowski was born in Warsaw, (t ...
, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others. A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.

# Definition

## Kuratowski closure operators and weakenings

Let $X$ be an arbitrary set and $\wp\left(X\right)$ its
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
. A Kuratowski closure operator is a
unary operation In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
$\mathbf:\wp\left(X\right) \to \wp\left(X\right)$ with the following properties: A consequence of $\mathbf$ preserving binary unions is the following condition: In fact if we rewrite the equality in 4'' as an inclusion, giving the weaker axiom 4'''' (''subadditivity''): then it is easy to see that axioms 4''' and 4'''' together are equivalent to 4'' (see the next-to-last paragraph of Proof 2 below). includes a fifth (optional) axiom requiring that singleton sets should be stable under closure: for all $x \in X$, $\mathbf\left(\\right) = \$. He refers to topological spaces which satisfy all five axioms as ''T1-spaces'' in contrast to the more general spaces which only satisfy the four listed axioms. Indeed, these spaces correspond exactly to the topological T1-spaces via the usual correspondence (see below).. If requirement 3'' is omitted, then the axioms define a Čech closure operator. If 1'' is omitted instead, then an operator satisfying 2'', 3'' and 4''' is said to be a Moore closure operator. A pair $\left(X, \mathbf\right)$ is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by $\mathbf$.

## Alternative axiomatizations

The four Kuratowski closure axioms can be replaced by a single condition, given by Pervin: Axioms 1''– 4'' can be derived as a consequence of this requirement: # Choose $A = B = \varnothing$. Then $\varnothing \cup \mathbf\left(\varnothing\right) \cup \mathbf\left(\mathbf\left(\varnothing\right)\right) = \mathbf\left(\varnothing\right) \setminus \mathbf\left(\varnothing\right) = \varnothing$, or $\mathbf\left(\varnothing\right) \cup \mathbf\left(\mathbf\left(\varnothing\right)\right) = \varnothing$. This immediately implies 1''. # Choose an arbitrary $A \subseteq X$ and $B = \varnothing$. Then, applying axiom 1'', $A \cup \mathbf\left(A\right) = \mathbf\left(A\right)$, implying 2''. # Choose $A = \varnothing$ and an arbitrary $B \subseteq X$. Then, applying axiom 1'', $\mathbf\left(\mathbf\left(B\right)\right) = \mathbf\left(B\right)$, which is 3''. # Choose arbitrary $A,B \subseteq X$. Applying axioms 1''– 3'', one derives 4''. Alternatively, had proposed a weaker axiom that only entails 2''– 4'': Requirement 1'' is independent of '' : indeed, if $X \neq \varnothing$, the operator $\mathbf^\star : \wp\left(X\right) \to \wp\left(X\right)$ defined by the constant assignment $A \mapsto \mathbf^\star\left(A\right) := X$ satisfies '' but does not preserve the empty set, since $\mathbf^\star\left(\varnothing\right) = X$. Notice that, by definition, any operator satisfying '' is a Moore closure operator. A more symmetric alternative to '' was also proven by M. O. Botelho and M. H. Teixeira to imply axioms 2''– 4'':.

# Analogous structures

## Interior, exterior and boundary operators

A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map $\mathbf : \wp\left(X\right) \to \wp\left(X\right)$ satisfying the following similar requirements:. For these operators, one can reach conclusions that are completely analogous to what was inferred for Kuratowski closures. For example, all Kuratowski interior operators are ''isotonic'', i.e. they satisfy 4''', and because of intensivity 2'', it is possible to weaken the equality in 3'' to a simple inclusion. The duality between Kuratowski closures and interiors is provided by the natural complement operator on $\wp\left(X\right)$, the map $\mathbf : \wp\left(X\right) \to \wp\left(X\right)$ sending $A \mapsto \mathbf\left(A\right):= X \setminus A$. This map is an orthocomplementation on the power set lattice, meaning it satisfies
De Morgan's laws In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...
: if $\mathcal$ is an arbitrary set of indices and $\_ \subseteq \wp\left(X\right)$, $\mathbf\left(\bigcup_ A_i\right) = \bigcap_ \mathbf(A_i), \qquad \mathbf\left(\bigcap_ A_i\right) = \bigcup_ \mathbf(A_i).$ By employing these laws, together with the defining properties of $\mathbf$, one can show that any Kuratowski interior induces a Kuratowski closure (and vice versa), via the defining relation $\mathbf := \mathbf$ (and $\mathbf := \mathbf$). Every result obtained concerning $\mathbf$ may be converted into a result concerning $\mathbf$ by employing these relations in conjunction with the properties of the orthocomplementation $\mathbf$. further provides analogous axioms for Kuratowski exterior operators and Kuratowski boundary operators, which also induce Kuratowski closures via the relations $\mathbf := \mathbf$ and $\mathbf\left(A\right):= A \cup \mathbf\left(A\right)$.

## Abstract operators

Notice that axioms 1''– 4'' may be adapted to define an ''abstract'' unary operation $\mathbf c : L \to L$ on a general bounded lattice $\left(L,\land,\lor,\mathbf 0, \mathbf 1\right)$, by formally substituting set-theoretic inclusion with the partial order associated to the lattice, set-theoretic union with the join operation, and set-theoretic intersections with the meet operation; similarly for axioms 1''– 4''. If the lattice is orthocomplemented, these two abstract operations induce one another in the usual way. Abstract closure or interior operators can be used to define a generalized topology on the lattice. Since neither unions nor the empty set appear in the requirement for a Moore closure operator, the definition may be adapted to define an abstract unary operator $\mathbf : S \to S$ on an arbitrary
poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
$S$.

# Connection to other axiomatizations of topology

## Induction of topology from closure

A closure operator naturally induces a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
as follows. Let $X$ be an arbitrary set. We shall say that a subset $C\subseteq X$ is closed with respect to a Kuratowski closure operator $\mathbf : \wp\left(X\right) \to \wp\left(X\right)$ if and only if it is a ''fixed point'' of said operator, or in other words it is ''stable under'' $\mathbf$, i.e. $\mathbf\left(C\right) = C$. The claim is that the family of all subsets of the total space that are complements of closed sets satisfies the three usual requirements for a topology, or equivalently, the family

## Induction of closure from topology

Conversely, given a family $\kappa$ satisfying axioms 1''– 3'', it is possible to construct a Kuratowski closure operator in the following way: if $A \in \wp\left(X\right)$ and $A^\uparrow = \$ is the inclusion upset of $A$, then $\mathbf_\kappa(A) := \bigcap_ B$ defines a Kuratowski closure operator $\mathbf_\kappa$ on $\wp\left(X\right)$. 1Since $\varnothing^\uparrow = \wp\left(X\right)$, $\mathbf_\kappa\left(\varnothing\right)$ reduces to the intersection of all sets in the family $\kappa$; but $\varnothing \in \kappa$ by axiom 1'', so the intersection collapses to the null set and 1'' follows. 2'' By definition of $A^\uparrow$, we have that $A \subseteq B$ for all $B \in \left\left(\kappa \cap A^\uparrow\right\right)$, and thus $A$ must be contained in the intersection of all such sets. Hence follows extensivity 2''. 3'' Notice that, for all $A \in \wp\left(X\right)$, the family $\mathbf_\kappa\left(A\right)^\uparrow \cap \kappa$ contains $\mathbf_\kappa\left(A\right)$ itself as a minimal element w.r.t. inclusion. Hence $\mathbf_\kappa^2(A) = \bigcap_B = \mathbf_\kappa(A)$, which is idempotence 3''. 4''' Let $A \subseteq B \subseteq X$: then $B^\uparrow \subseteq A^\uparrow$, and thus $\kappa \cap B^\uparrow \subseteq \kappa \cap A^\uparrow$. Since the latter family may contain more elements than the former, we find $\mathbf_\kappa\left(A\right) \subseteq \mathbf_\kappa\left(B\right)$, which is isotonicity 4'''. Notice that isotonicity implies $\mathbf_\kappa\left(A\right) \subseteq \mathbf_\kappa\left(A\cup B\right)$ and $\mathbf_\kappa\left(B\right) \subseteq \mathbf_\kappa\left(A\cup B\right)$, which together imply $\mathbf_\kappa\left(A\right) \cup \mathbf_\kappa\left(B\right) \subseteq \mathbf_\kappa\left(A\cup B\right)$. 4'' Finally, fix $A,B \in \wp\left(X\right)$. Axiom 2'' implies $\mathbf_\kappa\left(A\right), \mathbf_\kappa\left(B\right) \in \kappa$; furthermore, axiom 2'' implies that $\mathbf_\kappa\left(A\right) \cup \mathbf_\kappa\left(B\right) \in \kappa$. By extensivity 2'' one has $\mathbf_\kappa\left(A\right) \in A^\uparrow$ and $\mathbf_\kappa\left(B\right) \in B^\uparrow$, so that $\mathbf_\kappa\left(A\right) \cup \mathbf_\kappa\left(B\right) \in \left\left(A^\uparrow\right\right) \cap \left\left(B^\uparrow\right\right)$. But $\left\left(A^\uparrow\right\right) \cap \left\left(B^\uparrow\right\right) = \left(A \cup B\right)^\uparrow$, so that all in all $\mathbf_\kappa\left(A\right) \cup \mathbf_\kappa\left(B\right) \in \kappa\cap \left(A \cup B\right)^\uparrow$. Since then $\mathbf_\kappa\left(A \cup B\right)$ is a minimal element of $\kappa \cap \left(A \cup B\right)^\uparrow$ w.r.t. inclusion, we find $\mathbf_\kappa\left(A \cup B\right) \subseteq \mathbf_\kappa\left(A\right) \cup \mathbf_\kappa\left(B\right)$. Point 4. ensures additivity 4''.

## Exact correspondence between the two structures

In fact, these two complementary constructions are inverse to one another: if $\mathrm_\text\left(X\right)$ is the collection of all Kuratowski closure operators on $X$, and $\mathrm\left(X\right)$ is the collection of all families consisting of complements of all sets in a topology, i.e. the collection of all families satisfying 1''– 3'', then $\mathfrak : \mathrm_\text\left(X\right) \to \mathrm\left(X\right)$ such that

# __Examples_

*_As_discussed_above,_given_a_topological_space_$X$_we_may_define_the_closure_of_any_subset_$A_\subseteq_X$_to_be_the_set_$\mathbf\left(A\right)=\bigcap\$,_i.e._the_intersection_of_all_closed_sets_of_$X$_which_contain_$A$._The_set_$\mathbf\left(A\right)$_is_the_smallest_closed_set_of_$X$_containing_$A$,_and_the_operator_$\mathbf:\wp\left(X\right)_\to_\wp\left(X\right)$_is_a_Kuratowski_closure_operator. *_If_$X$_is_any_set,_the_operators_$\mathbf_\top,_\mathbf_\bot_:_\wp\left(X\right)_\to_\wp\left(X\right)$_such_that_\mathbf_\top(A)_=_\begin \varnothing_&_A_=_\varnothing,_\\ X_&_A_\neq_\varnothing, \end_\qquad_\mathbf_\bot(A)_=_A\quad_\forall_A_\in_\wp(X),are_Kuratowski_closures._The_first_induces_the_Trivial_topology.html" "title="kappa.html" ;"title="mathfrak[\kappa">mathfrak[\kappa is the family of all sets that are stable under $\mathbf_\kappa$, the result follows if both $\kappa\text{'} \subseteq \kappa$ and $\kappa \subseteq \kappa\text{'}$. Let $A \in \kappa\text{'}$: hence $\mathbf_\kappa\left(A\right) = A$. Since $\mathbf_\kappa\left(A\right)$ is the intersection of an arbitrary subfamily of $\kappa$, and the latter is complete under arbitrary intersections by 2'', then $A = \mathbf_\kappa\left(A\right) \in \kappa$. Conversely, if $A \in \kappa$, then $\mathbf_\kappa\left(A\right)$ is the minimal superset of $A$ that is contained in $\kappa$. But that is trivially $A$ itself, implying $A \in \kappa\text{'}$. We observe that one may also extend the bijection $\mathfrak$ to the collection $\mathrm_\left(X\right)$ of all Čech closure operators, which strictly contains $\mathrm_\text\left(X\right)$; this extension $\overline$ is also surjective, which signifies that all Čech closure operators on $X$ also induce a topology on $X$. However, this means that $\overline$ is no longer a bijection.

# Examples

* As discussed above, given a topological space $X$ we may define the closure of any subset $A \subseteq X$ to be the set $\mathbf\left(A\right)=\bigcap\$, i.e. the intersection of all closed sets of $X$ which contain $A$. The set $\mathbf\left(A\right)$ is the smallest closed set of $X$ containing $A$, and the operator $\mathbf:\wp\left(X\right) \to \wp\left(X\right)$ is a Kuratowski closure operator. * If $X$ is any set, the operators $\mathbf_\top, \mathbf_\bot : \wp\left(X\right) \to \wp\left(X\right)$ such that $\mathbf_\top(A) = \begin \varnothing & A = \varnothing, \\ X & A \neq \varnothing, \end \qquad \mathbf_\bot(A) = A\quad \forall A \in \wp(X),$are Kuratowski closures. The first induces the Trivial topology">indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...
$\$, while the second induces the discrete topology $\wp\left(X\right)$. *Fix an arbitrary $S \subsetneq X$, and let $\mathbf_S: \wp\left(X\right) \to \wp\left(X\right)$ be such that $\mathbf_S\left(A\right) := A \cup S$ for all $A \in \wp\left(X\right)$. Then $\mathbf_S$ defines a Kuratowski closure; the corresponding family of closed sets
cofinite topology In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocou ...
on $X$; if $\lambda = \aleph_1$, it induces the
cocountable topology The cocountable topology or countable complement topology on any set ''X'' consists of the empty set and all cocountable subsets of ''X'', that is all sets whose complement in ''X'' is countable. It follows that the only closed subsets are ''X'' and ...
.

# Properties

* Since any Kuratowski closure is isotonic, and so is obviously any inclusion mapping, one has the (isotonic)
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fun ...
$\langle \mathbf: \wp\left(X\right) \to \mathrm\left(\mathbf\right);\iota : \mathrm\left(\mathbf\right) \hookrightarrow \wp\left(X\right) \rangle$, provided one views $\wp\left(X\right)$as a poset with respect to inclusion, and $\mathrm\left(\mathbf\right)$ as a subposet of $\wp\left(X\right)$. Indeed, it can be easily verified that, for all $A \in \wp\left(X\right)$ and $C \in \mathrm\left(\mathbf\right)$, $\mathbf\left(A\right) \subseteq C$ if and only if $A \subseteq \iota\left(C\right)$. * If $\_$ is a subfamily of $\wp\left(X\right)$, then $\bigcup_ \mathbf(A_i) \subseteq \mathbf\left(\bigcup_ A_i\right), \qquad \mathbf\left(\bigcap_ A_i\right) \subseteq \bigcap_ \mathbf(A_i).$ * If $A,B \in \wp\left(X\right)$, then $\mathbf\left(A\right) \setminus \mathbf\left(B\right) \subseteq \mathbf\left(A\setminus B\right)$.

# Topological concepts in terms of closure

## Refinements and subspaces

A pair of Kuratowski closures $\mathbf_1, \mathbf_2 : \wp\left(X\right) \to \wp\left(X\right)$ such that $\mathbf_2\left(A\right) \subseteq \mathbf_1\left(A\right)$ for all $A \in \wp\left(X\right)$ induce topologies $\tau_1,\tau_2$ such that $\tau_1 \subseteq \tau_2$, and vice versa. In other words, $\mathbf_1$ dominates $\mathbf_2$ if and only if the topology induced by the latter is a refinement of the topology induced by the former, or equivalently

## Continuous maps, closed maps and homeomorphisms

A function $f:\left(X,\mathbf\right)\to \left(Y,\mathbf\text{'}\right)$ is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
at a point $p$ iff $p\in\mathbf\left(A\right) \Rightarrow f\left(p\right)\in\mathbf\text{'}\left(f\left(A\right)\right)$, and it is continuous everywhere iff $f(\mathbf(A)) \subseteq \mathbf'(f(A))$ for all subsets $A \in \wp\left(X\right)$. The mapping $f$ is a closed map iff the reverse inclusion holds, and it is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
iff it is both continuous and closed, i.e. iff equality holds.

## Separation axioms

Let $\left(X, \mathbf\right)$ be a Kuratowski closure space. Then * $X$ is a T0-space iff $x \neq y$ implies $\mathbf\left(\\right) \neq \mathbf\left(\\right)$; * $X$ is a T1-space iff $\mathbf\left(\\right)=\$ for all $x \in X$; * $X$ is a T2-space iff $x \neq y$ implies that there exists a set $A \in \wp\left(X\right)$ such that both $x \notin \mathbf\left(A\right)$ and $y \notin \mathbf\left(\mathbf\left(A\right)\right)$, where $\mathbf$ is the set complement operator.

## Closeness and separation

A point $p$ is
close Close may refer to: Music * ''Close'' (Kim Wilde album), 1988 * ''Close'' (Marvin Sapp album), 2017 * ''Close'' (Sean Bonniwell album), 1969 * "Close" (Sub Focus song), 2014 * "Close" (Nick Jonas song), 2016 * "Close" (Rae Sremmurd song), 201 ...
to a subset $A$ if $p\in\mathbf\left(A\right).$This can be used to define a proximity relation on the points and subsets of a set. Two sets $A,B \in \wp\left(X\right)$ are separated iff $\left(A \cap \mathbf\left(B\right)\right) \cup \left(B \cap \mathbf\left(A\right)\right) = \varnothing$. The space $X$ is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
iff it cannot be written as the union of two separated subsets..

* * * * * * *

# References

*. *. ** * . *. *{{Citation, last=Monteiro, first=António, title=Caractérisation de l'opération de fermeture par un seul axiome, date=September 1943, work=Portugaliae mathematica, volume=4, issue=4, pages=158–160, publication-date=1945, url=http://purl.pt/2135, trans-title=Characterization of the operation of closure by a single axiom, language=fr, zbl=0060.39406, author-link=Antonio Monteiro (mathematician).

# External links

Alternative Characterizations of Topological Spaces
Closure operators Mathematical axioms