In

'' '' (''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 $x\; \backslash in\; X$, $\backslash mathbf(\backslash )\; =\; \backslash $. He refers to topological spaces which satisfy all five axioms as ''T_{1}-spaces'' in contrast to the more general spaces which only satisfy the four listed axioms. Indeed, these spaces correspond exactly to the topological T_{1}-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 $(X,\; \backslash mathbf)$ is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by $\backslash mathbf$.

\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 $\backslash mathbf\_\backslash kappa$, the result follows if both $\backslash kappa\text{'}\; \backslash subseteq\; \backslash kappa$ and $\backslash kappa\; \backslash subseteq\; \backslash kappa\text{'}$. Let $A\; \backslash in\; \backslash kappa\text{'}$: hence $\backslash mathbf\_\backslash kappa(A)\; =\; A$. Since $\backslash mathbf\_\backslash kappa(A)$ is the intersection of an arbitrary subfamily of $\backslash kappa$, and the latter is complete under arbitrary intersections by 2'', then $A\; =\; \backslash mathbf\_\backslash kappa(A)\; \backslash in\; \backslash kappa$. Conversely, if $A\; \backslash in\; \backslash kappa$, then $\backslash mathbf\_\backslash kappa(A)$ is the minimal superset of $A$ that is contained in $\backslash kappa$. But that is trivially $A$ itself, implying $A\; \backslash in\; \backslash kappa\text{'}$.
We observe that one may also extend the bijection $\backslash mathfrak$ to the collection $\backslash mathrm\_(X)$ of all Čech closure operators, which strictly contains $\backslash mathrm\_\backslash text(X)$; this extension $\backslash overline$ is also surjective, which signifies that all Čech closure operators on $X$ also induce a topology on $X$. However, this means that $\backslash overline$ is no longer a bijection.

$\backslash $, while the second induces the discrete topology $\backslash wp(X)$.
*Fix an arbitrary $S\; \backslash subsetneq\; X$, and let $\backslash mathbf\_S:\; \backslash wp(X)\; \backslash to\; \backslash wp(X)$ be such that $\backslash mathbf\_S(A)\; :=\; A\; \backslash cup\; S$ for all $A\; \backslash in\; \backslash wp(X)$. Then $\backslash mathbf\_S$ defines a Kuratowski closure; the corresponding family of closed sets $\backslash mathfrak;\; href="/html/ALL/l/mathbf\_S.html"\; ;"title="mathbf\_S">mathbf\_S$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 $\backslash wp(X)$ itspower 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 ...

$\backslash mathbf:\backslash wp(X)\; \backslash to\; \backslash wp(X)$ with the following properties:
A consequence of $\backslash mathbf$ preserving binary unions is the following condition:
In fact if we rewrite the equality in 4'' as an inclusion, giving the weaker axiom ''">4Alternative 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 $A\; =\; B\; =\; \backslash varnothing$. Then $\backslash varnothing\; \backslash cup\; \backslash mathbf(\backslash varnothing)\; \backslash cup\; \backslash mathbf(\backslash mathbf(\backslash varnothing))\; =\; \backslash mathbf(\backslash varnothing)\; \backslash setminus\; \backslash mathbf(\backslash varnothing)\; =\; \backslash varnothing$, or $\backslash mathbf(\backslash varnothing)\; \backslash cup\; \backslash mathbf(\backslash mathbf(\backslash varnothing))\; =\; \backslash varnothing$. This immediately implies 1''. # Choose an arbitrary $A\; \backslash subseteq\; X$ and $B\; =\; \backslash varnothing$. Then, applying axiom 1'', $A\; \backslash cup\; \backslash mathbf(A)\; =\; \backslash mathbf(A)$, implying 2''. # Choose $A\; =\; \backslash varnothing$ and an arbitrary $B\; \backslash subseteq\; X$. Then, applying axiom 1'', $\backslash mathbf(\backslash mathbf(B))\; =\; \backslash mathbf(B)$, which is 3''. # Choose arbitrary $A,B\; \backslash subseteq\; X$. 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 $X\; \backslash neq\; \backslash varnothing$, the operator $\backslash mathbf^\backslash star\; :\; \backslash wp(X)\; \backslash to\; \backslash wp(X)$ defined by the constant assignment $A\; \backslash mapsto\; \backslash mathbf^\backslash star(A)\; :=\; X$ satisfies '' but does not preserve the empty set, since $\backslash mathbf^\backslash star(\backslash varnothing)\; =\; X$. 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 $\backslash mathbf\; :\; \backslash wp(X)\; \backslash to\; \backslash wp(X)$ 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 $\backslash wp(X)$, the map $\backslash mathbf\; :\; \backslash wp(X)\; \backslash to\; \backslash wp(X)$ sending $A\; \backslash mapsto\; \backslash mathbf(A):=\; X\; \backslash setminus\; A$. This map is an orthocomplementation on the power set lattice, meaning it satisfiesDe 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 $\backslash mathcal$ is an arbitrary set of indices and $\backslash \_\; \backslash subseteq\; \backslash wp(X)$,
$$\backslash mathbf\backslash left(\backslash bigcup\_\; A\_i\backslash right)\; =\; \backslash bigcap\_\; \backslash mathbf(A\_i),\; \backslash qquad\; \backslash mathbf\backslash left(\backslash bigcap\_\; A\_i\backslash right)\; =\; \backslash bigcup\_\; \backslash mathbf(A\_i).$$
By employing these laws, together with the defining properties of $\backslash mathbf$, one can show that any Kuratowski interior induces a Kuratowski closure (and vice versa), via the defining relation $\backslash mathbf\; :=\; \backslash mathbf$ (and $\backslash mathbf\; :=\; \backslash mathbf$). Every result obtained concerning $\backslash mathbf$ may be converted into a result concerning $\backslash mathbf$ by employing these relations in conjunction with the properties of the orthocomplementation $\backslash mathbf$.
further provides analogous axioms for Kuratowski exterior operators and Kuratowski boundary operators, which also induce Kuratowski closures via the relations $\backslash mathbf\; :=\; \backslash mathbf$ and $\backslash mathbf(A):=\; A\; \backslash cup\; \backslash mathbf(A)$.
Abstract operators

Notice that axioms 1''– 4'' may be adapted to define an ''abstract'' unary operation $\backslash mathbf\; c\; :\; L\; \backslash to\; L$ on a general bounded lattice $(L,\backslash land,\backslash lor,\backslash mathbf\; 0,\; \backslash mathbf\; 1)$, 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 $\backslash mathbf\; :\; S\; \backslash to\; S$ on an arbitraryposet
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 atopology
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 $C\backslash subseteq\; X$ is closed with respect to a Kuratowski closure operator $\backslash mathbf\; :\; \backslash wp(X)\; \backslash to\; \backslash wp(X)$ if and only if it is a ''fixed point'' of said operator, or in other words it is ''stable under'' $\backslash mathbf$, i.e. $\backslash mathbf(C)\; =\; C$. 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 $\backslash mathfrak;\; href="/html/ALL/l/mathbf.html"\; ;"title="mathbf">mathbf$Induction of closure from topology

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

In fact, these two complementary constructions are inverse to one another: if $\backslash mathrm\_\backslash text(X)$ is the collection of all Kuratowski closure operators on $X$, and $\backslash mathrm(X)$ 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 $\backslash mathfrak\; :\; \backslash mathrm\_\backslash text(X)\; \backslash to\; \backslash mathrm(X)$ such that $\backslash mathbf\; \backslash mapsto\; \backslash mathfrak;\; href="/html/ALL/l/mathbf.html"\; ;"title="mathbf">mathbf$__Examples_

*_As_discussed_above,_given_a_topological_space_$X$_we_may_define_the_closure_of_any_subset_$A\_\backslash subseteq\_X$_to_be_the_set_$\backslash mathbf(A)=\backslash bigcap\backslash $,_i.e._the_intersection_of_all_closed_sets_of_$X$_which_contain_$A$._The_set_$\backslash mathbf(A)$_is_the_smallest_closed_set_of_$X$_containing_$A$,_and_the_operator_$\backslash mathbf:\backslash wp(X)\_\backslash to\_\backslash wp(X)$_is_a_Kuratowski_closure_operator. *_If_$X$_is_any_set,_the_operators_$\backslash mathbf\_\backslash top,\_\backslash mathbf\_\backslash bot\_:\_\backslash wp(X)\_\backslash to\_\backslash wp(X)$_such_that_Examples

* As discussed above, given a topological space $X$ we may define the closure of any subset $A\; \backslash subseteq\; X$ to be the set $\backslash mathbf(A)=\backslash bigcap\backslash $, i.e. the intersection of all closed sets of $X$ which contain $A$. The set $\backslash mathbf(A)$ is the smallest closed set of $X$ containing $A$, and the operator $\backslash mathbf:\backslash wp(X)\; \backslash to\; \backslash wp(X)$ is a Kuratowski closure operator. * If $X$ is any set, the operators $\backslash mathbf\_\backslash top,\; \backslash mathbf\_\backslash bot\; :\; \backslash wp(X)\; \backslash to\; \backslash wp(X)$ such that $$\backslash mathbf\_\backslash top(A)\; =\; \backslash begin\; \backslash varnothing\; \&\; A\; =\; \backslash varnothing,\; \backslash \backslash \; X\; \&\; A\; \backslash neq\; \backslash varnothing,\; \backslash end\; \backslash qquad\; \backslash mathbf\_\backslash bot(A)\; =\; A\backslash quad\; \backslash forall\; A\; \backslash in\; \backslash wp(X),$$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 ...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 $\backslash lambda\; =\; \backslash aleph\_1$, 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 ...

$\backslash langle\; \backslash mathbf:\; \backslash wp(X)\; \backslash to\; \backslash mathrm(\backslash mathbf);\backslash iota\; :\; \backslash mathrm(\backslash mathbf)\; \backslash hookrightarrow\; \backslash wp(X)\; \backslash rangle$, provided one views $\backslash wp(X)$as a poset with respect to inclusion, and $\backslash mathrm(\backslash mathbf)$ as a subposet of $\backslash wp(X)$. Indeed, it can be easily verified that, for all $A\; \backslash in\; \backslash wp(X)$ and $C\; \backslash in\; \backslash mathrm(\backslash mathbf)$, $\backslash mathbf(A)\; \backslash subseteq\; C$ if and only if $A\; \backslash subseteq\; \backslash iota(C)$.
* If $\backslash \_$ is a subfamily of $\backslash wp(X)$, then $$\backslash bigcup\_\; \backslash mathbf(A\_i)\; \backslash subseteq\; \backslash mathbf\backslash left(\backslash bigcup\_\; A\_i\backslash right),\; \backslash qquad\; \backslash mathbf\backslash left(\backslash bigcap\_\; A\_i\backslash right)\; \backslash subseteq\; \backslash bigcap\_\; \backslash mathbf(A\_i).$$
* If $A,B\; \backslash in\; \backslash wp(X)$, then $\backslash mathbf(A)\; \backslash setminus\; \backslash mathbf(B)\; \backslash subseteq\; \backslash mathbf(A\backslash setminus\; B)$.
Topological concepts in terms of closure

Refinements and subspaces

A pair of Kuratowski closures $\backslash mathbf\_1,\; \backslash mathbf\_2\; :\; \backslash wp(X)\; \backslash to\; \backslash wp(X)$ such that $\backslash mathbf\_2(A)\; \backslash subseteq\; \backslash mathbf\_1(A)$ for all $A\; \backslash in\; \backslash wp(X)$ induce topologies $\backslash tau\_1,\backslash tau\_2$ such that $\backslash tau\_1\; \backslash subseteq\; \backslash tau\_2$, and vice versa. In other words, $\backslash mathbf\_1$ dominates $\backslash mathbf\_2$ if and only if the topology induced by the latter is a refinement of the topology induced by the former, or equivalently $\backslash mathfrak;\; href="/html/ALL/l/mathbf\_1.html"\; ;"title="mathbf\_1">mathbf\_1$Continuous maps, closed maps and homeomorphisms

A function $f:(X,\backslash mathbf)\backslash to\; (Y,\backslash mathbf\text{'})$ iscontinuous
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 $p\backslash in\backslash mathbf(A)\; \backslash Rightarrow\; f(p)\backslash in\backslash mathbf\text{'}(f(A))$, and it is continuous everywhere iff $$f(\backslash mathbf(A))\; \backslash subseteq\; \backslash mathbf\text{'}(f(A))$$ for all subsets $A\; \backslash in\; \backslash wp(X)$. 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 $(X,\; \backslash mathbf)$ be a Kuratowski closure space. Then * $X$ is a TCloseness and separation

A point $p$ isclose
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 $p\backslash in\backslash mathbf(A).$This can be used to define a proximity relation on the points and subsets of a set.
Two sets $A,B\; \backslash in\; \backslash wp(X)$ are separated iff $(A\; \backslash cap\; \backslash mathbf(B))\; \backslash cup\; (B\; \backslash cap\; \backslash mathbf(A))\; =\; \backslash varnothing$. 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..
See also

* * * * * * *Notes

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