From ABC-spaces to arbitrary spaces via fibrous preorders

Nelson Martins-Ferreira
Tuesday, July 23, 2013 - from 17:30 to 17:55
In modern terms, the main result in [1] establishes a categorical equivalence between preorders and A-spaces. A preorder is simply a reflexive and transitive relation while an A-space is a topological space in which any intersection of open sets is open. In [2], p.61, Ern\'e writes Hence the question arises: How can we enlarge the category of A-spaces on the one hand and the category of quasiordered sets on the other hand, so that we still keep an equivalence between the topological and the order-theoretical structures, but many interesting 'classical' topologies are included in the extended definition? and proposes the notions of B-spaces and C-spaces. With a different motivation in mind we will show that the category of fibrous preorders (to be introduced) is equivalent to the category of topological spaces. A fibrous preorder is a generalization of a preorder and was obtained as an attempt in finding a description for a topological space in terms of internal categorical structures. [1] Alexandrov, P., Diskrete Raume, Mat. Sbornik (N.S.) \textbf{2} (1937), 501--518. [2] Ern\'e, M., The ABC of order and topology, in: Herrlich, H., and Porst, H.-E. (eds), Category Theory at Work, Heldermann Verlag, Berlin, 1991, 57--83.