In the analysis of complex networks, the description of evolutionary
processes, or investigations into dynamics on fitness or energy landscapes notions such
as similarity, neighborhood, connectedness, or continuity of change appear in a natural
way. These concepts are of an inherently topological nature. Nevertheless, the
connection to the mathematical discipline of point set topology is rarely made in the
literature, presumably because in most applications there is no natural object
corresponding to an open or closed set. The link to textbook topology thus cannot be
made in a straightforward manner. Many of the deep results of point set topology still
remain valid, however, when open sets are abandoned and generalizations of the closure
operator are used as the foundation of the mathematical theory. Here we survey some
applications of such generalized point set topologies to chemistry and biology,
providing an overview of the underlying mathematical structures.
Keywords: Barrier tree, chemical reaction, closure, continuity, energy
landscape, evolution, fitness landscape, folding, graph, graph grammar,
hypergraph, neighborhood, network, RNA structure, similarity, topological
structures, topology.
