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.
This category Top embeds into Chu(Set, 2) due to the presence of a full and faithful functor F : Top → Chu(Set, 2), which assigns to each topological space (X, T) its equivalent representation (X, ∈, T).