Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that 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 formalized by Kazimierz Kuratowski,^[1] and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro,^[2] among others.

A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.^[3]

Let ${\displaystyle X}$ be an arbitrary set and ${\displaystyle \wp (X)}$ its power set. A Kuratowski closure operator is a unary operation ${\displaystyle \mathbf {c} :\wp (X)\to \wp (X)}$ with the following properties:

[K1] It preserves the empty set: ${\displaystyle \mathbf {c} (\varnothing )=\varnothing }$;

[K2] It is extensive: for all ${\displaystyle A\subseteq X}$, ${\displaystyle A\subseteq \mathbf {c} (A)}$;

[K3] It is idempotent: for all ${\displaystyle A\subseteq X}$, ${\displaystyle \mathbf {c} (A)=\mathbf {c} (\mathbf {c} (A))}$;

[K4] It preserves/distributes over binary unions: for all ${\displaystyle A,B\subseteq X}$, ${\displaystyle \mathbf {c} (A\cup B)=\mathbf {c} (A)\cup \mathbf {c} (B)}$.

A consequence of ${\displaystyle \mathbf {c} }$ preserving binary unions is the following condition:^[4]

[K4'] It is monotone: ${\displaystyle A\subseteq B\Rightarrow \mathbf {c} (A)\subseteq \mathbf {c} (B)}$.

In fact if we rewrite the equality in [K4] as an inclusion, giving the weaker axiom [K4''] (subadditivity):

[K4''] It is subadditive: for all ${\displaystyle A,B\subseteq X}$, ${\displaystyle \mathbf {c} (A\cup B)\subseteq \mathbf {c} (A)\cup \mathbf {c} (B)}$,

then it is easy to see that axioms [K4'] and [K4''] together are equivalent to [K4] (see the next-to-last paragraph of Proof 2 below).

Kuratowski (1966) includes a fifth (optional) axiom requiring that singleton sets should be stable under closure: for all ${\displaystyle x\in X}$, ${\displaystyle \mathbf {c} (\{x\})=\{x\}}$. 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).^[5]

If requirement [K3] is omitted, then the axioms define a Čech closure operator.^[6] If [K1] is omitted instead, then an operator satisfying [K2], [K3] and [K4'] is said to be a Moore closure operator.^[7] A pair ${\displaystyle (X,\mathbf {c} )}$ is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by ${\displaystyle \mathbf {c} }$.

The four Kuratowski closure axioms can be replaced by a single condition, given by Pervin:^[8]

[P] For all ${\displaystyle A,B\subseteq X}$, ${\displaystyle A\cup \mathbf {c} (A)\cup \mathbf {c} (\mathbf {c} (B))=\mathbf {c} (A\cup B)\setminus \mathbf {c} (\varnothing )}$.

Axioms [K1]–[K4] can be derived as a consequence of this requirement:

1. Choose ${\displaystyle A=B=\varnothing }$. Then ${\displaystyle \varnothing \cup \mathbf {c} (\varnothing )\cup \mathbf {c} (\mathbf {c} (\varnothing ))=\mathbf {c} (\varnothing )\setminus \mathbf {c} (\varnothing )=\varnothing }$, or ${\displaystyle \mathbf {c} (\varnothing )\cup \mathbf {c} (\mathbf {c} (\varnothing ))=\varnothing }$. This immediately implies [K1].
2. Choose an arbitrary ${\displaystyle A\subseteq X}$ and ${\displaystyle B=\varnothing }$. Then, applying axiom [K1], ${\displaystyle A\cup \mathbf {c} (A)=\mathbf {c} (A)}$, implying [K2].
3. Choose ${\displaystyle A=\varnothing }$ and an arbitrary ${\displaystyle B\subseteq X}$. Then, applying axiom [K1], ${\displaystyle \mathbf {c} (\mathbf {c} (B))=\mathbf {c} (B)}$, which is [K3].
4. Choose arbitrary ${\displaystyle A,B\subseteq X}$. Applying axioms [K1]–[K3], one derives [K4].

Alternatively, Monteiro (1945) had proposed a weaker axiom that only entails [K2]–[K4]:^[9]

[M] For all ${\displaystyle A,B\subseteq X}$, ${\textstyle A\cup \mathbf {c} (A)\cup \mathbf {c} (\mathbf {c} (B))\subseteq \mathbf {c} (A\cup B)}$.

Requirement [K1] is independent of [M] : indeed, if ${\displaystyle X\neq \varnothing }$, the operator ${\displaystyle \mathbf {c} ^{\star }:\wp (X)\to \wp (X)}$ defined by the constant assignment ${\displaystyle A\mapsto \mathbf {c} ^{\star }(A):=X}$ satisfies [M] but does not preserve the empty set, since ${\displaystyle \mathbf {c} ^{\star }(\varnothing )=X}$. Notice that, by definition, any operator satisfying [M] is a Moore closure operator.

A more symmetric alternative to [M] was also proven by M. O. Botelho and M. H. Teixeira to imply axioms [K2]–[K4]:^[2]

[BT] For all ${\displaystyle A,B\subseteq X}$, ${\textstyle A\cup B\cup \mathbf {c} (\mathbf {c} (A))\cup \mathbf {c} (\mathbf {c} (B))=\mathbf {c} (A\cup B)}$.

A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map ${\displaystyle \mathbf {i} :\wp (X)\to \wp (X)}$ satisfying the following similar requirements:^[3]

[I1] It preserves the total space: ${\displaystyle \mathbf {i} (X)=X}$;

[I2] It is intensive: for all ${\displaystyle A\subseteq X}$, ${\displaystyle \mathbf {i} (A)\subseteq A}$;

[I3] It is idempotent: for all ${\displaystyle A\subseteq X}$, ${\displaystyle \mathbf {i} (\mathbf {i} (A))=\mathbf {i} (A)}$;

[I4] It preserves binary intersections: for all ${\displaystyle A,B\subseteq X}$, ${\displaystyle \mathbf {i} (A\cap B)=\mathbf {i} (A)\cap \mathbf {i} (B)}$.

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 [K4'], and because of intensivity [I2], it is possible to weaken the equality in [I3] to a simple inclusion.

The duality between Kuratowski closures and interiors is provided by the natural complement operator on ${\displaystyle \wp (X)}$, the map ${\displaystyle \mathbf {n} :\wp (X)\to \wp (X)}$ sending ${\displaystyle A\mapsto \mathbf {n} (A):=X\setminus A}$. This map is an orthocomplementation on the power set lattice, meaning it satisfies De Morgan's laws: if ${\displaystyle {\mathcal {I}}}$ is an arbitrary set of indices and ${\displaystyle \{A_{i}\}_{i\in {\mathcal {I}}}\subseteq \wp (X)}$, $${\displaystyle \mathbf {n} \left(\bigcup _{i\in {\mathcal {I}}}A_{i}\right)=\bigcap _{i\in {\mathcal {I}}}\mathbf {n} (A_{i}),\qquad \mathbf {n} \left(\bigcap _{i\in {\mathcal {I}}}A_{i}\right)=\bigcup _{i\in {\mathcal {I}}}\mathbf {n} (A_{i}).}$$

By employing these laws, together with the defining properties of ${\displaystyle \mathbf {n} }$, one can show that any Kuratowski interior induces a Kuratowski closure (and vice versa), via the defining relation ${\displaystyle \mathbf {c} :=\mathbf {nin} }$ (and ${\displaystyle \mathbf {i} :=\mathbf {ncn} }$). Every result obtained concerning ${\displaystyle \mathbf {c} }$ may be converted into a result concerning ${\displaystyle \mathbf {i} }$ by employing these relations in conjunction with the properties of the orthocomplementation ${\displaystyle \mathbf {n} }$.

Pervin (1964) further provides analogous axioms for Kuratowski exterior operators^[3] and Kuratowski boundary operators,^[10] which also induce Kuratowski closures via the relations ${\displaystyle \mathbf {c} :=\mathbf {ne} }$ and ${\displaystyle \mathbf {c} (A):=A\cup \mathbf {b} (A)}$.

Notice that axioms [K1]–[K4] may be adapted to define an abstract unary operation ${\displaystyle \mathbf {c} :L\to L}$ on a general bounded lattice ${\displaystyle (L,\land ,\lor ,\mathbf {0} ,\mathbf {1} )}$, 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 [I1]–[I4]. 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 ${\displaystyle \mathbf {c} :S\to S}$ on an arbitrary poset ${\displaystyle S}$.

A closure operator naturally induces a topology as follows. Let ${\displaystyle X}$ be an arbitrary set. We shall say that a subset ${\displaystyle C\subseteq X}$ is closed with respect to a Kuratowski closure operator ${\displaystyle \mathbf {c} :\wp (X)\to \wp (X)}$ if and only if it is a fixed point of said operator, or in other words it is stable under ${\displaystyle \mathbf {c} }$, i.e. ${\displaystyle \mathbf {c} (C)=C}$. 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 ${\displaystyle {\mathfrak {S}}[\mathbf {c} ]}$ of all closed sets satisfies the following:

[T1] It is a bounded sublattice of ${\displaystyle \wp (X)}$, i.e. ${\displaystyle X,\varnothing \in {\mathfrak {S}}[\mathbf {c} ]}$;

[T2] It is complete under arbitrary intersections, i.e. if ${\displaystyle {\mathcal {I}}}$ is an arbitrary set of indices and ${\displaystyle \{C_{i}\}_{i\in {\mathcal {I}}}\subseteq {\mathfrak {S}}[\mathbf {c} ]}$, then ${\textstyle \bigcap _{i\in {\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]}$;

[T3] It is complete under finite unions, i.e. if ${\displaystyle {\mathcal {I}}}$ is a finite set of indices and ${\displaystyle \{C_{i}\}_{i\in {\mathcal {I}}}\subseteq {\mathfrak {S}}[\mathbf {c} ]}$, then ${\textstyle \bigcup _{i\in {\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]}$.

Notice that, by idempotency [K3], one may succinctly write ${\displaystyle {\mathfrak {S}}[\mathbf {c} ]=\operatorname {im} (\mathbf {c} )}$.

Conversely, given a family ${\displaystyle \kappa }$ satisfying axioms [T1]–[T3], it is possible to construct a Kuratowski closure operator in the following way: if ${\displaystyle A\in \wp (X)}$ and ${\displaystyle A^{\uparrow }=\{B\in \wp (X)\ |\ A\subseteq B\}}$ is the inclusion upset of ${\displaystyle A}$, then $${\displaystyle \mathbf {c} _{\kappa }(A):=\bigcap _{B\in (\kappa \cap A^{\uparrow })}B}$$

defines a Kuratowski closure operator ${\displaystyle \mathbf {c} _{\kappa }}$ on ${\displaystyle \wp (X)}$.

In fact, these two complementary constructions are inverse to one another: if ${\displaystyle \mathrm {Cls} _{\text{K}}(X)}$ is the collection of all Kuratowski closure operators on ${\displaystyle X}$, and ${\displaystyle \mathrm {Atp} (X)}$ is the collection of all families consisting of complements of all sets in a topology, i.e. the collection of all families satisfying [T1]–[T3], then ${\displaystyle {\mathfrak {S}}:\mathrm {Cls} _{\text{K}}(X)\to \mathrm {Atp} (X)}$ such that ${\displaystyle \mathbf {c} \mapsto {\mathfrak {S}}[\mathbf {c} ]}$ is a bijection, whose inverse is given by the assignment ${\displaystyle {\mathfrak {C}}:\kappa \mapsto \mathbf {c} _{\kappa }}$.

We observe that one may also extend the bijection ${\displaystyle {\mathfrak {S}}}$ to the collection ${\displaystyle \mathrm {Cls} _{\check {C}}(X)}$ of all Čech closure operators, which strictly contains ${\displaystyle \mathrm {Cls} _{\text{K}}(X)}$; this extension ${\displaystyle {\overline {\mathfrak {S}}}}$ is also surjective, which signifies that all Čech closure operators on ${\displaystyle X}$ also induce a topology on ${\displaystyle X}$.^[11] However, this means that ${\displaystyle {\overline {\mathfrak {S}}}}$ is no longer a bijection.

• As discussed above, given a topological space ${\displaystyle X}$ we may define the closure of any subset ${\displaystyle A\subseteq X}$ to be the set ${\displaystyle \mathbf {c} (A)=\bigcap \{C{\text{ a closed subset of }}X|A\subseteq C\}}$, i.e. the intersection of all closed sets of ${\displaystyle X}$ which contain ${\displaystyle A}$. The set ${\displaystyle \mathbf {c} (A)}$ is the smallest closed set of ${\displaystyle X}$ containing ${\displaystyle A}$, and the operator ${\displaystyle \mathbf {c} :\wp (X)\to \wp (X)}$ is a Kuratowski closure operator.
• If ${\displaystyle X}$ is any set, the operators ${\displaystyle \mathbf {c} _{\top },\mathbf {c} _{\bot }:\wp (X)\to \wp (X)}$ such that $${\displaystyle \mathbf {c} _{\top }(A)={\begin{cases}\varnothing &A=\varnothing ,\\X&A\neq \varnothing ,\end{cases}}\qquad \mathbf {c} _{\bot }(A)=A\quad \forall A\in \wp (X),}$$are Kuratowski closures. The first induces the indiscrete topology ${\displaystyle \{\varnothing ,X\}}$, while the second induces the discrete topology ${\displaystyle \wp (X)}$.

• Fix an arbitrary ${\displaystyle S\subsetneq X}$, and let ${\displaystyle \mathbf {c} _{S}:\wp (X)\to \wp (X)}$ be such that ${\displaystyle \mathbf {c} _{S}(A):=A\cup S}$ for all ${\displaystyle A\in \wp (X)}$. Then ${\displaystyle \mathbf {c} _{S}}$ defines a Kuratowski closure; the corresponding family of closed sets ${\displaystyle {\mathfrak {S}}[\mathbf {c} _{S}]}$ coincides with ${\displaystyle S^{\uparrow }}$, the family of all subsets that contain ${\displaystyle S}$. When ${\displaystyle S=\varnothing }$, we once again retrieve the discrete topology ${\displaystyle \wp (X)}$ (i.e. ${\displaystyle \mathbf {c} _{\varnothing }=\mathbf {c} _{\bot }}$, as can be seen from the definitions).
• If ${\displaystyle \lambda }$ is an infinite cardinal number such that ${\displaystyle \lambda \leq \operatorname {crd} (X)}$, then the operator ${\displaystyle \mathbf {c} _{\lambda }:\wp (X)\to \wp (X)}$ such that$${\displaystyle \mathbf {c} _{\lambda }(A)={\begin{cases}A&\operatorname {crd} (A)<\lambda ,\\X&\operatorname {crd} (A)\geq \lambda \end{cases}}}$$satisfies all four Kuratowski axioms.^[12] If ${\displaystyle \lambda =\aleph _{0}}$, this operator induces the cofinite topology on ${\displaystyle X}$; if ${\displaystyle \lambda =\aleph _{1}}$, it induces the cocountable topology.

• Since any Kuratowski closure is isotonic, and so is obviously any inclusion mapping, one has the (isotonic) Galois connection ${\displaystyle \langle \mathbf {c} :\wp (X)\to \mathrm {im} (\mathbf {c} );\iota :\mathrm {im} (\mathbf {c} )\hookrightarrow \wp (X)\rangle }$, provided one views ${\displaystyle \wp (X)}$as a poset with respect to inclusion, and ${\displaystyle \mathrm {im} (\mathbf {c} )}$ as a subposet of ${\displaystyle \wp (X)}$. Indeed, it can be easily verified that, for all ${\displaystyle A\in \wp (X)}$ and ${\displaystyle C\in \mathrm {im} (\mathbf {c} )}$, ${\displaystyle \mathbf {c} (A)\subseteq C}$ if and only if ${\displaystyle A\subseteq \iota (C)}$.
• If ${\displaystyle \{A_{i}\}_{i\in {\mathcal {I}}}}$ is a subfamily of ${\displaystyle \wp (X)}$, then $${\displaystyle \bigcup _{i\in {\mathcal {I}}}\mathbf {c} (A_{i})\subseteq \mathbf {c} \left(\bigcup _{i\in {\mathcal {I}}}A_{i}\right),\qquad \mathbf {c} \left(\bigcap _{i\in {\mathcal {I}}}A_{i}\right)\subseteq \bigcap _{i\in {\mathcal {I}}}\mathbf {c} (A_{i}).}$$
• If ${\displaystyle A,B\in \wp (X)}$, then ${\displaystyle \mathbf {c} (A)\setminus \mathbf {c} (B)\subseteq \mathbf {c} (A\setminus B)}$.

A pair of Kuratowski closures ${\displaystyle \mathbf {c} _{1},\mathbf {c} _{2}:\wp (X)\to \wp (X)}$ such that ${\displaystyle \mathbf {c} _{2}(A)\subseteq \mathbf {c} _{1}(A)}$ for all ${\displaystyle A\in \wp (X)}$ induce topologies ${\displaystyle \tau _{1},\tau _{2}}$ such that ${\displaystyle \tau _{1}\subseteq \tau _{2}}$, and vice versa. In other words, ${\displaystyle \mathbf {c} _{1}}$ dominates ${\displaystyle \mathbf {c} _{2}}$ if and only if the topology induced by the latter is a refinement of the topology induced by the former, or equivalently ${\displaystyle {\mathfrak {S}}[\mathbf {c} _{1}]\subseteq {\mathfrak {S}}[\mathbf {c} _{2}]}$.^[13] For example, ${\displaystyle \mathbf {c} _{\top }}$ clearly dominates ${\displaystyle \mathbf {c} _{\bot }}$(the latter just being the identity on ${\displaystyle \wp (X)}$). Since the same conclusion can be reached substituting ${\displaystyle \tau _{i}}$ with the family ${\displaystyle \kappa _{i}}$ containing the complements of all its members, if ${\displaystyle \mathrm {Cls} _{\text{K}}(X)}$ is endowed with the partial order ${\displaystyle \mathbf {c} \leq \mathbf {c} '\iff \mathbf {c} (A)\subseteq \mathbf {c} '(A)}$ for all ${\displaystyle A\in \wp (X)}$ and ${\displaystyle \mathrm {Atp} (X)}$ is endowed with the refinement order, then we may conclude that ${\displaystyle {\mathfrak {S}}}$ is an antitonic mapping between posets.

In any induced topology (relative to the subset A) the closed sets induce a new closure operator that is just the original closure operator restricted to A: ${\displaystyle \mathbf {c} _{A}(B)=A\cap \mathbf {c} _{X}(B)}$, for all ${\displaystyle B\subseteq A}$.^[14]

A function ${\displaystyle f:(X,\mathbf {c} )\to (Y,\mathbf {c} ')}$ is continuous at a point ${\displaystyle p}$ iff ${\displaystyle p\in \mathbf {c} (A)\Rightarrow f(p)\in \mathbf {c} '(f(A))}$, and it is continuous everywhere iff $${\displaystyle f(\mathbf {c} (A))\subseteq \mathbf {c} '(f(A))}$$ for all subsets ${\displaystyle A\in \wp (X)}$.^[15] The mapping ${\displaystyle f}$ is a closed map iff the reverse inclusion holds,^[16] and it is a homeomorphism iff it is both continuous and closed, i.e. iff equality holds.^[17]

Let ${\displaystyle (X,\mathbf {c} )}$ be a Kuratowski closure space. Then

• ${\displaystyle X}$ is a T₀-space iff ${\displaystyle x\neq y}$ implies ${\displaystyle \mathbf {c} (\{x\})\neq \mathbf {c} (\{y\})}$;^[18]
• ${\displaystyle X}$ is a T₁-space iff ${\displaystyle \mathbf {c} (\{x\})=\{x\}}$ for all ${\displaystyle x\in X}$;^[19]
• ${\displaystyle X}$ is a T₂-space iff ${\displaystyle x\neq y}$ implies that there exists a set ${\displaystyle A\in \wp (X)}$ such that both ${\displaystyle x\notin \mathbf {c} (A)}$ and ${\displaystyle y\notin \mathbf {c} (\mathbf {n} (A))}$, where ${\displaystyle \mathbf {n} }$ is the set complement operator.^[20]

A point ${\displaystyle p}$ is close to a subset ${\displaystyle A}$ if ${\displaystyle p\in \mathbf {c} (A).}$This can be used to define a proximity relation on the points and subsets of a set.^[21]

Two sets ${\displaystyle A,B\in \wp (X)}$ are separated iff ${\displaystyle (A\cap \mathbf {c} (B))\cup (B\cap \mathbf {c} (A))=\varnothing }$. The space ${\displaystyle X}$ is connected iff it cannot be written as the union of two separated subsets.^[22]

• Characterizations of the category of topological spaces
• Čech closure operator – Closure operatorPages displaying short descriptions of redirect targets
• Closure operator – Mathematical operator
• Closure algebra – Algebraic structure
• Kuratowski's closure-complement problem
• Preclosure operator – Closure operator
• Pretopological space – Generalized topological space
• Topological space – Mathematical space with a notion of closeness

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• Kuratowski, Kazimierz (1966) [1958], Topology, vol. I, translated by Jaworowski, J., Academic Press, ISBN 0-12-429201-1, LCCN 66029221.
  • —— (2010). "On the Operation Ā Analysis Situs". ResearchGate. Translated by Mark Bowron.
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• Monteiro, António (1945), "Caractérisation de l'opération de fermeture par un seul axiome" [Characterization of the operation of closure by a single axiom], Portugaliae mathematica (in French), vol. 4, no. 4, pp. 158–160, MR 0012310, Zbl 0060.39406.

• Alternative Characterizations of Topological Spaces