Analele Ştiinţifice ale Universităţii 'Al. I. Cuza' din Iasi. Secţiunea I a. Matematică. Serie Nouă (An. Şti. Univ. 'Al. I. Cuza'' Iaşi Secţ. I a Mat. (N.S.)) (19750101), 21, 15-24. ISSN: 0041-9109 (print).
Subject
54 General topology -- 54A Generalities 54A15 Syntopogeneous structures
The author defines an approximation structure to be a triple $\{X,\scr A,A\}$, where $X$ and $A$ are two sets and $\scr A$ is a subset of $X\times\scr R(A,X)$, $\scr R(A,X)$ denotes the set of all subsets of $A\times X$, and $\scr A$ is assumed to satisfy the condition (a) for $x\in X$ there exists $\alpha\in\scr R(A,X)$ such that $(x,\alpha)\in\scr A$. The terminology is motivated by some analogies in computational mathematics. For this kind of structure, initial structures and, more generally, projective limits are defined. If $\scr A$ satisfies the condition (a) and $M\subset X$, set $\scr I_{\scr A}(M)=$ \{$y\in X\colon(y,\alpha)\in\scr A$ for $S(\alpha)=M$\}, where $S(\alpha)=$ \{$y\in X\colon(a,y)\in\alpha$ for $a\in A$\}. The connection between these set-valued set functions $\scr I_{\scr A}\colon\scr P(X)\rightarrow\scr P(X)$ and those considered by P. C. Hammer [Nieuw Arch. Wisk. (3) {\bf 10} (1962), 55--77; MR0176433 (31 \#705)] is discussed. Further, set $\scr A[x]=\{\alpha\in\scr R(A,X)\colon(x,\alpha)\in\scr A\}$ for $x\in X$, and $\scr A[M]=$ \{$\alpha\in\scr R(A,X)\colon\alpha\in\scr A[x]$ for $x\in M$\} for $M\subset X$. Now assume that $\scr A$ satisfies the condition (a) described above and the conditions (b) for $M\subset X$, $\alpha\in\scr A[M]$, $a\in A$, we have $M\subset r(\alpha,a)=\{z\in X\colon(a,z)\in\alpha\}$, and (c) $\scr A[X]\neq\varnothing$. For $a\in A$ and $M,N\subset X$, define $M<_aN$ if there is $\alpha\in\scr A[M]$ such that $r(\alpha,a)\subset N$. Then $<_a$ is a semitopogenous order in the sense of the reviewer [English translation, {\it Foundations of general topology}, Pergamon, Oxford, 1963; MR0157340 (28 \#575)]. Conditions are given for these $<_a$ to constitute a syntopogenous structure, and it is shown that every syntopogenous structure is obtained in this way.