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.