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I was looking again at equilogical space; I wasnâ€™t able to determine the importance of the $T_0$ condition either from the page or from looking over the article Equilogical Spaces by Bauer, Birkedal, and Scott. (As best I can tell, it might be because they want to study the connection with algebraic lattices, as opposed to some preorder analogue of algebraic lattices.) From the point of view of desirable properties of the category of equilogical spaces, is there any particular advantage to using $T_0$-spaces as opposed to arbitrary spaces?
Not as far as I know.
I added some comments to equilogical space about their relationship to the exact completion of Top.
Perhaps Dana Scott did want the equivalence relation to relate to the topology to the extent that topologically equivalent points would be equivalent. Then he realised that (assuming choice) the category of such spaces was equivalent to its full subcategory of $T_0$ spaces, and this description was simpler (or at least more familiar).
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