The category Top of topological spaces and their continuous functions embeds in Chu(Set, 2) in the sense that there exists a full and faithful functor F : Top → Chu(Set, 2) providing for each topological space (X, T) its representation F((X, T)) = (X, ∈, T) as noted above.
The category of topological spaces and their continuous functions is contained within the larger category Chu(Set, 2), as evidenced by the existence of a full and faithful functor F that associates each space (X, T) with its corresponding representation (X, ∈, T).