It was already known that the category of T0 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 FILc C FILb C FILa in the literature. We establish adjunctions between FILa and ASSM and between FILc and ASSM, and show that FILb and FILc are equivalent to reflective full subcategories of ASSM. The corresponding categories FILb0 and FILc0 of T0 spaces are equivalent to full subcategories of EQU.
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Reinhold Heckmann /