The galaxies of nonstandard enlargements of infinite and transfinite graphs

The galaxies of the nonstandard enlargements of connected, conventionally infinite graphs as well as of walk-connected transfinite graphs are defined, analyzed, and illustrated by some examples. It is then shown that any such enlargement either has exactly one galaxy, its principal one, or it has infinitely many galaxies. In the latter case, the galaxies are partially ordered by their “closeness” to the principal galaxy. If an enlargement has a galaxy ΓΓ different from its principal galaxy, then it has a two-way infinite sequence of galaxies that contains ΓΓ and is totally ordered according to that “closeness” property. There may be many such totally ordered sequences.Furthermore, a walk-connected graph G1G1 of transfinite rank 1 consists in general of connected conventional graphs (graphs of rank 0, called 0-sections) that are walk-connected together at their infinite extremities. The enlargement ∗G1 of G1G1 consists of the enlargement of G1G1, as well as of the enlargements of its 0-sections. The latter enlargements are all contained within the principal galaxy of ∗G1. Moreover, ∗G1 may have other galaxies of rank 1; these too are partially and totally ordered as before. These results extend to the enlargements of transfinite graphs of ranks greater than 1.