On the Relationship between Filter Spaces and Equilogical Spaces

It was already known that the category of T 0 topological spaces is not itself cartesian closed, but can be embedded into the cartesian closed categories FIL of filter spaces and EQU of equilogical spaces where the latter embeds into the cartesian closed category ASSM of assemblies over algebraic lattices. Here, we first clarify the notion of filter space---there are at least three versions FIL a ' FIL b ' FIL c in the literature. We establish adjunctions between FIL a and ASSM and between FIL c and ASSM, and show that FIL b and FIL c are equivalent to reflective full subcategories of ASSM. The corresponding categories FIL b 0 and FIL c 0 of T 0 spaces are equivalent to full subcategories of EQU. Keywords: Categorical models and logics, domain theory and applications Author's address: Reinhold Heckmann, FB 14 -- Informatik, Universitat des Saarlandes, Postfach 151150, D-66041 Saarbrucken, Germany Phone: +49 681 302 2454 Fax: +49 681 302 3065 e-mail: heckmann@cs.un..