This paper investigates the interplay between properties of a topological space \( X \), in particular of its natural order, and properties of the lax comma category \( \mathsf{Top}\Downarrow X \), where \( \mathsf{Top} \) denotes the category of topological spaces and continuous maps. Namely, it is shown that, whenever \( X \) is a topological \( \bigwedge \)-semilattice, the canonical forgetful functor \( \mathsf{Top}\Downarrow X \to \mathsf{Top} \) is topological, preserves and reflects exponentials, and preserves effective descent morphisms. Moreover, under additional conditions on \( X \), a characterisation of effective descent morphisms is obtained.