On a Lax-Algebraic Characterization of Closed Maps

M. M. Clementino and W. Tholen presented recently a lax-algebraic generalization of the well-known result on the coincidence of two classes of continuous maps between topological spaces: proper (maps, whose pullbacks are closed) and perfect (closed maps with compact fibres). Their achievement depends on a particular lax-algebraic extension of the concept of closed map, which relies on constructively completely distributive quantales. This paper proposes a definition of closed maps, which is not dependant on the underlying quantale of lax algebras, and proves the results of M. M. Clementino and W. Tholen in the new setting. We also show that our presented notion fits the axiomatic definition of closed maps in a category.