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  • An important example of a frame is the lattice of

    2021-10-27

    An important example of a frame is the lattice of open subsets of any topological space X. The correspondence is clearly functorial (by taking inverse images), and consequently one has a contravariant functor with the category of topological spaces and continuous maps as domain category. The functor has a right adjoint, the spectrum functor, which assigns to each frame L its spectrum ΣL, that is, the space of all homomorphisms with open sets for any , and to each frame homomorphism the continuous map such that . The category and its dual category of locales and localic maps are the framework of pointfree topology. The contravariance of the functor indicates that on the extension step from classical topology into pointfree topology, it is in that the classical notions have to be considered. For instance, the generalized pointfree subspaces are subobjects in (the sublocales), that is, quotients in . A map of locales (localic map) is the unique right adjoint of a frame homomorphism . They are precisely the maps that, besides preserving arbitrary meets, reflect the top Estradiol valerate (i.e., ) and satisfy for every and . The reader is referred to [19] for more information on frames and locales. Regarding the frame of reals mentioned at the Introduction, it is worth pointing out that the assignments determine a surjective frame homomorphism from to (the usual topology on the rationals) as it obviously turns the defining relations (r1)–(r6) into identities in . Consequently, for every . Moreover, if and only if , and if and only if . For any frame L, a continuous real-valued function[4] (resp. extended continuous real-valued function[6]) on a frame L is a frame homomorphism (resp. ). We denote by and , respectively, the collections of all continuous real-valued functions and extended continuous real-valued functions on L. For each , we denote by the constant function defined by if and otherwise. There is a useful way of specifying (extended) continuous real-valued functions on a frame L with the help of scales ([12, Section 4]). An extended scale in L is a map such that whenever . An extended scale is a scale if For each extended scale σ in L, the formulas determine an ([6, Lemma 1]); then, if and only if σ is a scale. For more about continuous real-valued functions on frames we refer to [4].
    Metric hedgehog frames Consider , the hedgehog frame with κ spines which was introduced earlier. It is obvious that (since condition (h0) is vacuously satisfied in this case). Moreover, is also isomorphic to . The isomorphism is induced by the following correspondences (where φ denotes any increasing bijection between and ): We now introduce the following notation in : The set forms a base for (since it is closed under finite meets by (h3) and (h4)). Hence for any .
    The spatial reflection of a frame L is the unit map of the adjunction , that is, the frame homomorphism In the present case, as seen in the previous proof, the homeomorphism induces an isomorphism mapping to and to . Hence, we have the following result:
    Recall that the weight ([10]) of a frame L is the smallest infinite cardinal for which there exists a base B for L of cardinality .
    It may be worth pointing out that the formula from [21, Theorem 4.6] provides the expression for the metric diameter on induced by the metric uniformity. Of course, this is a totally bounded metric frame if and only if .
    A uniform frame L is said to be complete whenever any dense surjection of uniform frames is a frame isomorphism. In the proof of next result we shall make use of the following well-known fact about surjective frame homomorphisms : (One first notes that in M iff , since Then, by regularity, , which shows that h is one-one.)
    Of course, in order to specify a continuous hedgehog-valued function on L, we only need to define it on the generators of and to check that it turns the conditions (h0)–(h4) into identities in L.