Directed sets and topological spaces definable in o‐minimal structures

We study directed sets definable in o-minimal structures, showing that expansions of ordered fields these admit cofinal curves, as well a suitable analogue groups, and furthermore no holds full generality. use the theory tame pairs to extend results field case families with finite intersection property. then apply our topologies. prove all topological spaces display properties akin first countability, give several characterizations notion compactness due Peterzil Steinhorn generalized this setting.