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Applied Categorical Structures

, Volume 27, Issue 6, pp 687–702 | Cite as

Maximal Lindelöf Locales

  • Themba DubeEmail author
Article
  • 37 Downloads

Abstract

For a subfit frame L, let \({\mathcal {S}}_{\mathfrak {c}}(L)\) denote the complete Boolean algebra whose elements are the sublocales of L that are joins of closed sublocales. Identifying every element of L with the open sublocale it determines allows us to view L as a subframe of \({\mathcal {S}}_{\mathfrak {c}}(L)\). With this backdrop, we say L is maximal Lindelöf if it is Lindelöf and whenever \(L\subseteq M\), for some Lindelöf subframe M of \({\mathcal {S}}_{\mathfrak {c}}(L)\), then \(L=M\). Recall that a topological space \((X,\tau )\) is maximal Lindelöf if it is Lindelöf, and there is no strictly finer topology \(\rho \) on X such that \((X,\rho )\) is a Lindelöf space. We show that a space is maximal Lindelöf if and only if the frame of its open subsets is maximal Lindelöf. We then characterize maximal Lindelöf frames internally. Among regular frames, we show that the maximal Lindelöf frames are precisely the Lindelöf ones in which every \(F_\sigma \)-sublocale (meaning a join of countably many closed sublocales) is closed.

Keywords

Frame Subframe Locale Sublocale Maximal-Lindelöf frame Localic map Closed localic map Open localic map 

Mathematics Subject Classification

Primary 06D22 Secondary 54A10 54D20 

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Notes

Acknowledgements

The author is greatly indebted to the referee for pointing out some crucial omissions and also for several helpful comments that resulted in a much improved paper.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of South AfricaPretoriaSouth Africa

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