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More on Normality and Related Properties

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Separation in Point-Free Topology

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Abstract

Here we start with two more variants of normality. There is the perfect normality, which turns out to be a conjunction of the classical perfectness (which is slightly different in the point-free context due to the different behaviour of sublocales and subspaces) and normality; in a way it can be viewed as a weaker form of metrizability. Next we deal with the technically important collectionwise normality. Then, in the penultimate section we prove and discuss the Katětov–Tong insertion theorem, using (to advantage) the techniques of the point-free real line. We finish with a certain duality between normality and extremal disconnectedness that allows to translate several results concerning normality to facts about extremal disconnected frames.

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Notes

  1. 1.

    In [68]; see also Gilmour [114].

  2. 2.

    These are the point-free extensions of the κ-normal spaces introduced by Schepin in 1972 in [244] (and introduced as mildly normal spaces in [263]). Clearly, a space X is κ-normal if and only if Ω(X) is mildly normal.

  3. 3.

    The σ-frame ϱL consists precisely of all countable joins of regular elements of L since any finite meet of regular elements is regular by (4) in A.2.1.1.

  4. 4.

    Two subsets U and V  of X are separable if there is a continuous map such that f[U] = {1} and f[V ] = {0}; of course, this is only possible if the closures of U and V  are disjoint.

  5. 5.

    Originally announced by Tong in 1948 (the proof was, however, not published until 1952 [273]), Katětov shares the name of the theorem because of his independent version of 1951, with a simpler proof [177]. Such results have roots in Baire (1905, [8]), Hahn (1917, [134]) and Dieudonné (1944, [78]) who proved it for the real line, metrizable spaces and paracompact spaces, respectively.

  6. 6.

    Katětov relations appear in the literature under various other names such as, e.g. quasi-proximities or subordinations. They share some of the defining properties of the so-called strong relations that describe the compactifications of a frame [27].

  7. 7.

    T. Kubiak was the first to notice that several pairs of results in classical topology characterizing the concepts of normality and extremal disconnectedness show a “remarkable duality” [187] between the two concepts: each pair is identical in structure but prove facts about normal spaces on one side of the pair and about extremally disconnected spaces on the other [185]. The point-free study of this duality was undertaken in [131].

  8. 8.

    F-frames are the frames in which the open quotient of any (dense) cozero element is a C ∗-quotient.

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Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal

    Jorge Picado

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

    Aleš Pultr

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Picado, J., Pultr, A. (2021). More on Normality and Related Properties. In: Separation in Point-Free Topology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-53479-0_8

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