On a topology and limits for inductive systems of $C^*$-algebras over partially ordered sets

Motivated by algebraic quantum field theory and our previous work we study properties of inductive systems of \ $C^*$-algebras over arbitrary partially ordered sets. A partially ordered set can be represented as the union of the family of its maximal upward directed subsets indexed by elements of a certain set. We consider a topology on the set of indices generated by a base of neighbourhoods. Examples of those topologies with different properties are given. An inductive system of $C^*$-algebras and its inductive limit arise naturally over each maximal upward directed subset. Using those inductive limits, we construct different types of $C^*$-algebras. In particular, for neighbourhoods of the topology on the set of indices we deal with the $C^*$-algebras which are the direct products of those inductive limits. The present paper is concerned with the above-mentioned topology and the algebras arising from an inductive system of $C^*$-algebras over a partially ordered set. We show that there exists a connection between properties of that topology and those $C^*$-algebras.