A lattice-theoretic approach to arbitrary real functions on frames
n this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if L is a subfit frame, arbitrary extended real functions on L are the elements of the Dedekind-MacNeille completion of the poset of all extended semicontinuo...
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| Format: | article |
| Publication Date: | 2017 |
| Country: | España |
| Institution: | Universidad del País Vasco |
| Repository: | Addi. Archivo Digital para la Docencia y la Investigación |
| OAI Identifier: | oai:addi.ehu.eus:10810/71920 |
| Online Access: | http://hdl.handle.net/10810/71920 |
| Access Level: | Open access |
| Keyword: | frame locale frame of reals continuous real function order complete Dedekind-MacNeille completion semicontinuous real function partial real function Hausdorff continuous real function |
| Summary: | n this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if L is a subfit frame, arbitrary extended real functions on L are the elements of the Dedekind-MacNeille completion of the poset of all extended semicontinuous functions on L. This approach mimicks the situation one has with a T1-space X, where the lattice F(X) of arbitrary extended real functions on X is the smallest complete lattice containing both extended upper and lower semicontinuous functions on X. Then, we identify real-valued functions by lattice-theoretic means. By construction, we obtain definitions of discontinuous functions that are conservative for T1-spaces. We also analyze semicontinuity and introduce definitions which are conservative for T0-spaces. |
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