Reachability, Connectivity, and Proximity in Chemical Spaces

A chemical universe consists of a set X of chemical compounds and a set of reactions between them. Each reaction transforms a finite (small) multiset of educts in a small multiset of products. In the topological context explored here, stoichiometry is neglected, and products and educts are treated as simple sets. Reactions thus form directed hyperedges on X. Since X can be infinite, it is of interest to consider X not only for a combinatorial point of view but also as a topological construct. Here we argue that generalized reaches and relative closure functions provide a natural framework. These can be seen as generalizations of connected components and are equivalent to a certain class of separation or proximity spaces. We consider notions of strong and weak connected components and derive their basic properties, and we characterize the conditions under which they are equivalent to generalized closure spaces; as it turns out, chemical universes are very different from this more well-behaved type of generalized topologies. The theory presented here provides a solid ground to further investigate concepts related to connectivity in a very general class of models that in particular includes chemistry.