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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 6th 2024

    Created:

    Definition

    The dissolution locale L\mathfrak{C}L of a locale LL is defined as the poset of its sublocales (equivalently: nuclei on LL) equipped with the relation of reverse inclusion.

    There is a canonical morphism of locales

    ι:LL\iota\colon\mathfrak{C}L \to L

    such that the map ι *\iota^* sends an open aLa\in L to the open in L\mathfrak{C}L given by the open sublocale of aa.

    Interpretation

    The map LL\mathfrak{C}L\to L can be considered an analogue of the canonical map T dTT_d \to T for a topological space TT, where T dT_d is the underlying set of TT equipped with the discrete topology.

    In particular, discontinuous maps LML\to M could be defined as morphisms of locales LM\mathfrak{C}L\to M, see Picado–Pultr, XIV.7.3.

    References

    Original reference:

    • John R. Isbell, On dissolute spaces, Topology and its Applications 40:1 (1991), 63–70. doi.

    Expository account:

    • Frames and Locales, see Sections III.3, VI.4-6, and others. The dissolution frame is denoted there by 𝒮𝓁(L) op\mathcal{Sl}(L)^{op} (III.3.2) or by (L)\mathfrak{C}(L) (III.5.2) and the dissolution locale is denoted by 𝔖(L)\mathfrak{S}(L) (XIV.7.2).

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