Helly and Klee type intersection theorems for finitary connected paved spaces (Q1886722)

This is an extensive paper on abstract convexity. Every nonvoid subfamily of the family of all subsets of a set \(S \not = \emptyset\) is called a paving. In Part 1 many kinds of pavings are defined. For instance, a paving \(\mathcal P\) is called \(\cap_ f\)-closed provided \(A \in \mathcal P\) and \(B\in \mathcal P\) imply \(A \cap B \in \mathcal P\), \(\cup_ f\)-closed provided \(A \in \mathcal P\) and \(B\in \mathcal P\) imply \(A \cup B \in \mathcal P\), a lattice when it is \(\cap _ f\)-closed and \(\cup_ f\)-closed, coherent if for arbitrary sets \(A \in \mathcal P\) and \(B \in \mathcal P\) with nonvoid intersection we have \(A \cup B \in \mathcal P\). Also notions of compact paving, upward filtrated paving and an alignment are introduced. A large number of relationships between those and also other definitions are shown. Special attention is paid to the notions of the so-called \(n\)-ary connected paving (we omit here the rather long definition) and an abstract notion of separation. In Part 2 the so-called segment space is considered. Part 3 of this paper is devoted to Helly-type and Klee-type intersection theorems. Part 4 is about topological connectedness.