On completeness (Q5905394)

The author gives a general notion of completeness and a general construction of completion which generalizes both completion of metric spaces by Cauchy sequences and completion of ordered sets by Dedekind cuts. His construction is following: A basic system is a triple \({\mathfrak X}=(X,{\mathcal C},\Psi)\) where \(X\) is a set, \(\mathcal C\) is a system of subsets of \(X\) (elements of \(\mathcal C\) are called regions) and \(\Psi=(\psi_ n)\) is a sequence of mappings from \({\mathcal C}\) to \({\mathcal C}\) such that \(\psi_ n(A)\subseteq A\) for any \(A\in {\mathcal C}\) and \(n\in {\mathbb{N}}\). Such a system is called complete if every sequence \((A_ n)\) of regions for which the sequence \((\psi_ n(A_ n))\) is descending, has a nonempty intersection. A sequence \((E_ n)\) of regions is called regular if \(E_{n+1}\subseteq\psi_ n(E_ n)\) for any \(n\in{\mathbb{N}}\); a regular sequence converges to a point \(x\in X\) if its intersection is \(\{x\}\). A basic system \({\mathfrak X}\) is point regular, if every point in \(X\) is the limit of a regular sequence of regions and if the following holds: if \((E_ n)\) is a regular sequence of regions converging to \(x\) and \(A\) is a region containing \(x\) then \(E_ m\subseteq A\) for some \(m\in{\mathbb{N}}\). A system \({\mathfrak X}\) is idempotent if each mapping \(\psi_ n\) is idempotent. Two sequences \((E_ n)\), \((F_ n)\) of regions are called interlaced if any \(E_ n\) is subset of a suitable \(F_ m\) and vice versa. For a given basic system \({\mathfrak X}=(X,{\mathcal C},\Psi)\) let us identify regular sequences of regions which are interlaced and let \(X^*\) be the set of corresponding equivalence classes. For \(A\in{\mathcal C}\) let \(A^*\) be the set of all equivalence classes \(x^*\in X^*\) such that for any \((E_ n)\in x^*\) there exists \(m\in{\mathbb{N}}\) with \(E_ m\subseteq A\) and let \({\mathcal C}^*=\{A^*;A\in{\mathcal C}\}\). Further, for \(A^*\in {\mathcal C}^*\) put \(\psi^*_ n(A^*)=(\psi_ n(A))^*\) and \(\Psi^*=(\psi^*_ n)\). Then \({\mathfrak X}^*=(X^*,{\mathcal C}^*,\Psi^*)\) is a basic system. The main result is following: If \({\mathfrak X}=(X,{\mathcal C},\Psi)\) is a point regular, idempotent basic system then \({\mathfrak X}^*=(X^*,{\mathcal C}^*,\Psi^*)\) is a complete basic system; moreover, \({\mathfrak X}^*\) is a completion of \(\mathfrak X\) in the sense that there exists an injection \(i: X\rightarrow X^*\) such that \(i(A)\subseteq A^*\) for any \(A\in{\mathcal C}\). The author further shows that by special choice of a basic system his construction gives the completion of metric spaces and the completion of ordered sets.