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Localic real functions: A general setting by Javier Gutiérrez García; Tomasz Kubiak; Jorge Picado is a Mathematics article available to read on EtoBox.

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In pointfree topology the lattice-ordered ring of all continuous real functions on a frame L has not been a part of the lattice of all lower (or upper) semicontinuous real functions on L just because all those continuities involve different domains. This paper demonstrates a framework in which all those continuous and semicontinuous functions arise (up to isomorphism) as members of the lattice-ordered ring of all frame homomorphisms from the frame L(R) of reals into S(L), the dual of the co-frame of all sublocales of L. The latticeordered ring Frm(L(R), S(L)) is a pointfree counterpart of the ring R X with X a topological space. We thus have a pointfree analogue of the concept of an arbitrary not necessarily (semi) continuous real function on L. One feature of this remarkable conception is that one eventually has: lower semicontinuous + upper semicontinuous = continuous. We document its importance by showing how nicely can the insertion, extension and regularization theorems, proved earlier by these authors, be recast in the new setting.

Who reads Localic real functions: A general setting?

Author

Javier Gutiérrez García; Tomasz Kubiak; Jorge Picado

Publisher

Elsevier Science; Elsevier ; Elsevier BV (ISSN 0022-4049)

Published

2009

Language

EN

Field

Mathematics (Physical Sciences)