Weak geodesic topology and fixed finite subgraph theorems in infinite partial cubes I. Topologies and the geodesic convexity

The weak geodesic topology on the vertex set of a partial cube G is the finest weak topology on V ( G ) endowed with the geodesic convexity. We prove the equivalence of the following properties: (i) the space V ( G ) is compact; (ii) V ( G ) is weakly countably compact; (iii) the vertex set of any ray of G has a limit point; (iv) any concentrated subset of V ( G ) (i.e. a set A such that any two infinite subsets of A cannot be separated by deleting finitely many vertices) has a finite positive number of limit points. Moreover, if V ( G ) is compact, then it is scattered. We characterize the partial cubes for which the weak geodesic topology and the geodesic topology (see [N. Polat, Graphs without isometric rays and invariant subgraph properties I. J. Graph Theory27 (1998), 99–109]) coincide, and we show that the class of these particular partial cubes is closed under Cartesian products, retracts and gated amalgams.