A lattice-theoretic approach to arbitrary real functions on frames
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913668277469184 |
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| author | Carollo, Imanol Mozo |
| author_facet | Carollo, Imanol Mozo |
| contents | In 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 $T_1$-space $X$, where the lattice $\overline{\mathrm{F}}(X)$ 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 $T_1$-spaces. We also analyze semicontinuity and introduce definitions which are conservative for $T_D$-spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A lattice-theoretic approach to arbitrary real functions on frames Carollo, Imanol Mozo General Topology 06D22 (Primary) 26A15, 54C30, 54D15 (Secondary) In 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 $T_1$-space $X$, where the lattice $\overline{\mathrm{F}}(X)$ 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 $T_1$-spaces. We also analyze semicontinuity and introduce definitions which are conservative for $T_D$-spaces. |
| title | A lattice-theoretic approach to arbitrary real functions on frames |
| topic | General Topology 06D22 (Primary) 26A15, 54C30, 54D15 (Secondary) |
| url | https://arxiv.org/abs/2501.16833 |