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  • SKI II br Introduction Topological hedgehogs keep generating

    2022-09-29


    Introduction Topological hedgehogs keep generating interest in point-set topology as they are a rich source of counterexamples and applications (for a comprehensive survey on topological hedgehogs we refer to [1]; see also [9]). They may be described as a set of spines identified at a single point. Specifically, for each set I of cardinality κ, the classical metric hedgehog is the disjoint union of κ copies (the spines) of the real unit interval identified at the origin, with the topology generated by the metric given by (see e.g. [1, pp. 28] or [9, pp. 251]) One of the differences between point-set topology and pointfree topology is that one may present frames by generators and relations (similarly to the presentation of groups by generators and relations). Then, for a frame L defined by generators and relations one may define a morphism with domain L just by specifying its values on the generators; it SKI II is a frame homomorphism precisely when it turns the defining relations of L into identities in the codomain frame. In this paper we present the frame of the metric hedgehog, by specifying its generators and relations. This is done just from the rationals, independently of any notion of real number. For that we need to recall first that the frame of reals (see e.g. [4]) is the frame specified by generators for and defining relations Equivalently, can be specified (see [18]) by generators and for , subject to relations By dropping relations (r5) and (r6) one has the frame of extended reals ([6]). We introduce the metric hedgehog frame as a cardinal generalization of . Specifically, let κ be some cardinal and let I be a set of cardinality κ. The frame of the metric hedgehog with κ spines is the frame presented by generators and for and , subject to the defining relations The purpose of this paper is to present some of the main properties of the metric hedgehog frame (that from now on we shall mostly refer to as simply the hedgehog frame), as well as of the corresponding continuous hedgehog-valued functions. We prove that for each cardinal κ, the hedgehog frame is a metric frame of weight , complete in its metric uniformity. Then we show that the countable coproduct of the hedgehog frame with κ spines is universal in the class of metric frames of weight , that is, every metrizable frame of weight is embeddable into a countable cartesian power of the hedgehog frame. Being the hedgehog frame a fundamental example of a collectionwise normal frame, we take the opportunity to study collectionwise normality in frames, a concept originally introduced by A. Pultr ([22]) in connection with metrizability. First, we show that collectionwise normality is hereditary with respect to -sublocales and that it is a property invariant under closed maps. Then we present the counterparts of Urysohn's separation and Tietze's extension theorems for continuous hedgehog-valued functions. They both characterize κ-collectionwise normality.
    Preliminaries and notation A frame (or locale) L is a complete lattice (with bottom 0 and top 1) such that for all and . A frame is precisely a complete Heyting algebra with Heyting operation → satisfying the standard equivalence iff . The pseudocomplement of an is the element . A frame L is regular if, for each , where means that . A subset B of a frame L is a base for L if each element in L is join-generated by some set of elements in B. Given a base B for L, the pseudocomplement of is obviously given by . A frame homomorphism is a map between frames which preserves finitary meets (including the top element 1) and arbitrary joins (including the bottom element 0). Note that for every . A frame homomorphism h is said to be dense if implies and it is codense if implies . We denote by the category of frames and frame homomorphisms. As a frame homomorphism preserves arbitrary joins, it has a (unique) right adjoint determined by . In particular, and . The frame homomorphism h is a surjection if and only if is an embedding if and only if .