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Notes on Point-Free Topology

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New Perspectives in Algebra, Topology and Categories

Part of the book series: Coimbra Mathematical Texts ((CMT,volume 1))

Abstract

Point-free topology is the study of the category of locales and localic maps and its dual category of frames and frame homomorphisms. These notes cover the topics presented by the first author in his course on Frames and Locales at the Summer School in Algebra and Topology. We give an overview of the basic ideas and motivation for point-free topology, explaining the similarities and dissimilarities with the classical setting and stressing some of the new features.

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Notes

  1. 1.

    In fact this property is characteristic for the representation of continuous maps: if the spaces are sober, which is a very general condition (see 3.4 below), each frame homomorphism \(h:\Omega (Y)\rightarrow \Omega (X)\) is an \(\Omega (f)\) for some continuous f [49].

  2. 2.

    That is, the opens \(U=\mathrm{int}\,\overline{U}\), “open sets without lesions”, the open sets one thinks about first.

  3. 3.

    This proof shows indeed more: L/R is a sublocale of L, see 5.4 below.

  4. 4.

    Nuclei in L are in a one-one correspondence with onto frame homomorphisms with domain L hence constitute an alternative representation for sublocales in L [44].

  5. 5.

    The joins \(\mathop {\textstyle \bigsqcup }\) in S are given by \(\mathop {\textstyle \bigsqcup }s_i=\nu _S(\mathop {\textstyle \bigvee }s_i)\) (if \(t\in S\) and \(t\ge s_i\) for all i then \(t\ge \mathop {\textstyle \bigvee }s_i\) and \(t=\nu _S(t)\ge \nu _S(\mathop {\textstyle \bigvee }s_i)\)).

  6. 6.

    The reader might have expected for the definition of \(\mathfrak {o}(a)\). This subset of L is not a sublocale, but the intuition is not wide from the target: is isomorphic to \(\mathfrak o(a)\) which is the image of the localic map adjoint to the map .

  7. 7.

    The importance of this condition is comparable with that of sobriety. Note that in a way these two conditions are dual to each other: while sobriety requires that we cannot add a point to X without changing \(\Omega (X)\), \(T_D\) says that we cannot subtract a point.

  8. 8.

    More precisely: a map is localic iff each closed sublocale has a closed preimage whose complement is contained in the preimage of the complement of the original sublocale, and the least subocales are preserved.

  9. 9.

    With the Boolean Ultrafilter Theorem; the compactness of products for general spaces is equivalent with the full Axiom of Choice. Even the theorem for general frames is choice-free, but this is technically much more involved [5, 27, 37].

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Acknowledgements

The authors gratefully acknowledge financial support from the Centre for Mathematics of the University of Coimbra (UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES) and from the Department of Applied Mathematics (KAM) of Charles University (Prague). The first author also acknowledges the UCL project Attractivité internationale et collaborations de recherche dans le cadre du Coimbra group 2017–2020 and an ERASMUS+ Staff Mobility Grant from the University of Coimbra that supported his visit to the Université catholique de Louvain.

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Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    Jorge Picado

  2. Department of Applied Mathematics and ITI, MFF, Charles University, Prague, Czech Republic

    Aleš Pultr

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  1. Jorge Picado
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Correspondence to Jorge Picado .

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  1. Departamento de Matematica, Universidade de Coimbra, Coimbra, Portugal

    Maria Manuel Clementino

  2. Dipartimento di Matematica, University of Padova, Padua, Italy

    Alberto Facchini

  3. Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Marino Gran

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Picado, J., Pultr, A. (2021). Notes on Point-Free Topology. In: Clementino, M.M., Facchini, A., Gran, M. (eds) New Perspectives in Algebra, Topology and Categories. Coimbra Mathematical Texts, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-030-84319-9_6

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