closure axioms

A closure operator on a set $X$ is an operator which assigns a set $A^{c}$ to each subset $A$ of $X$ , and such that the following (Kuratowski’s closure axioms) hold for any subsets $A$ and $B$ of $X$ :

1.
$\\emptyset^{c}=\\emptyset$ ;
2.
$A\\subset A^{c}$ ;
3.
$(A^{c})^{c}=A^{c}$ ;
4.
$(A\\cup B)^{c}=A^{c}\\cup B^{c}.$

The following theorem due to Kuratowski says that a closure operator characterizes a unique topology on $X$ :

Theorem. Let $c$ be a closure operator on $X$ , and let $\\mathcal{T}=\\{X-A:A\\subseteq X,\\;A^{c}=A\\}$ . Then $\\mathcal{T}$ is a topology on $X$ , and $A^{c}$ is the $\\mathcal{T}$ -closure of $A$ for each subset $A$ of $X$ .

单词	ClosureAxioms
释义	closure axioms A closure operator on a set $X$ is an operator which assigns a set $A^{c}$ to each subset $A$ of $X$ , and such that the following (Kuratowski’s closure axioms) hold for any subsets $A$ and $B$ of $X$ : 1. $\\emptyset^{c}=\\emptyset$ ; 2. $A\\subset A^{c}$ ; 3. $(A^{c})^{c}=A^{c}$ ; 4. $(A\\cup B)^{c}=A^{c}\\cup B^{c}.$ The following theorem due to Kuratowski says that a closure operator characterizes a unique topology on $X$ : Theorem. Let $c$ be a closure operator on $X$ , and let $\\mathcal{T}=\\{X-A:A\\subseteq X,\\;A^{c}=A\\}$ . Then $\\mathcal{T}$ is a topology on $X$ , and $A^{c}$ is the $\\mathcal{T}$ -closure of $A$ for each subset $A$ of $X$ .
随便看	adherent point AdHoc ad hoc AdjacencyMatrix adjacency matrix Adjacent adjacent adjacent Adjacent1 AdjacentFraction adjacent fraction AdjoiningAnIdentityToASemigroup adjoining an identity to a semigroup Adjoint adjoint AdjointEndomorphism adjoint endomorphism AdjointRepresentation adjoint representation Adjugate adjugate Adjunctions AdjunctionSpace adjunction space Admissibility