  We have defined and established a theory of cofinite connectedness of a
cofinite graph. Many of the properties of connectedness of topological spaces
have analogs for cofinite connectedness. We have seen that if $G$ is a cofinite
group and Gamma=Gamma(G,X) is the Cayley graph. Then Gamma can be given a
suitable cofinite uniform topological structure so that $X$ generates $G$,
topologically iff Gamma is cofinitely connected.
  Our immediate next concern is developing group actions on cofinite graphs.
Defining the action of an abstract group over a cofinite graph in the most
natural way we are able to characterize a unique way of uniformizing an
abstract group with a cofinite structure, obtained from the cofinite structure
of the graph in the underlying action, so that the aforesaid action becomes
uniformly continuous.
