Format of Original
University of Houston
Houston Journal of Mathematics
Let X be a topological graph, with arcs joined only at end points. If Y is any locally connected metrizable compactum that is co-elementarily equivalent to X, then Y is homeomorphic to X. In particular, X and Y are homeomorphic if some lattice base for one is elementarily equivalent to some lattice base for the other.
Bankston, Paul, "Categoricity and Topological Graphs" (2012). Mathematics, Statistics and Computer Science Faculty Research and Publications. 98.