


Scott approach spaces



Description: 
Approach spaces, introduced by R. Lowen in 1989, constitute a generalization of both topological and metric spaces. There are several ways of describing them, among the most common are the one using distances between points and sets, and the one using numerified convergence which tells how far a filter is from converging to a point. In this talk we will consider approach spaces via regular function frames, i.e. a set of functions from X to [0,infty] which generalize the notion of closed set. We will define the corresponding regular function coframe (of opens). Unlike the case of topological spaces, these structures are not simply dual to each other; however, they do define equivalent categories. In 2006 B. Banaschewski, R. Lowen and C. Van Olmen describe the category AFrm of approach frames which plays in this context the role that frames play in Topology. We introduce the category BFrm, which should be seen as approach coframes rather than coapproach frames, and we prove that AFrm and BFrm are equivalent. This method of generalizing frames also indicates how to generalize other order structures to metric spaces, as for instance continuous lattices. Using regular function (co)frames it is also possible to define a Scott approach structure associated to any generalized metric space.

Date: 
20110329

Start Time: 
14:30 
Speaker: 
Gonçalo Gutierres (Mat. FCTUC/CMUC)

Institution: 
Mat. FCTUC/CMUC

Place: 
Sala 5.5

Research Groups: 
Algebra, Logic and Topology

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