Cell structures and completely metrizable spaces and their mappings
A (combinatorial) graph is a discrete set of vertices together with a set of edges. We define cell structures as inverse sequences of graphs with mild convergence conditions and we define cell mappings between cell structures. These cell structures yield completely metrizable spaces as perfect images of closed subsets of countable products of discrete spaces. Cell mappings between cell structures define the continuous mappings between the corresponding spaces. In this way we can envision a continuous mapping between metric spaces as the limit of a sequence of discrete approximations. Thus, cell structures provide a kind of bridge between discrete and continuous mathematics.