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The category of measurable spaces are topological over $Set$ in that they support initial & final structures similarly to that topological spaces.

A measurable space is a set supporting $\sigma$ - algebras. If we forget the $\sigma$ structure, and so only have closure under finite meets & unions does the category remain topological over $Set$.

It seems to me that they should since the lattice of such algebras on a space is complete.

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Are you asking whether the category of measurable spaces is topological over the category of (distributive) lattices? Or, that the finitely measurable spaces is topological over $Set$? –  Berci Feb 12 '13 at 14:50
Over $Set$. I'll edit the question. –  Mozibur Ullah Feb 12 '13 at 15:07

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