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The notions of closedness and \(D\)-connectedness in quantale-valued approach spaces. (English) Zbl 1450.54002
Approach spaces were introduced by Lowen in 1989. They are intrinsically linked to the Lawvere quantale \(([0,\infty], \geq, +)\) with bottom \(\infty\) and top \(0\). The paper deals with generalisations of approach spaces, where the quantale \(([0,\infty], \geq, +)\) is replaced by some quantale \((\mathcal{L}, \leq, *)\), satisfying certain supplementary conditions as studied by G. Jäger, by Lai and Tholen and others. The main purpose of the paper is to study lower separation axioms and \(D\)-connectedness in the setting of \(\mathcal{L}\)-approach gauge spaces.
First we make a general remark. Although an approach space \(X\) can be isomorphically described for instance by a distance \(\partial: X \times 2^X \to [0,\infty]\) or by a gauge \(\mathcal{G}\) of quasi-metrics \(d : X \times X \to [0,\infty]\), it is well known that for a general quantale \((\mathcal{L}, \leq, *)\), both generalisations, \(\mathcal{L}\)-approach gauges and \(\mathcal{L}\)-approach distances, need not induce isomorphic categories. Nevertheless the authors consider the notion of an \(\mathcal{L}\)-approach space \(X\), which can either be described by its \(\mathcal{L}\)-approach gauge or by its \(\mathcal{L}\)-approach distance. This is meaningless. When given an \(\mathcal{L}\)-approach gauge one can consider an associated \(\mathcal{L}\)-approach distance, but both structures do not uniquely determine each other. To formulate Theorems 3.8, 3.10 and 4.7 one should consider extra conditions on \(\mathcal{L}\) to ensure that the categories of \(\mathcal{L}\)-approach gauges and of \(\mathcal{L}\)-approach distances are isomorphic.
Besides the previous observation, Theorems 3.8, 3.10 and 4.7 and also 4.6 in the paper are false, even in the case of the Lawvere quantale \(([0,\infty], \geq, +)\) where the problem we just discussed vanishes. We use the interpretation of the theorems for the Lawvere quantale \(([0,\infty], \geq, +)\) and present some counterexamples in that setting.
In Theorem 3.8. it is claimed that for an approach space \(X\) with gauge \(\mathcal{G}\) and its related distance \(\partial\) conditions (ii) and (iii) are equivalent.
(ii)
\(x \not = p\) implies there exists \(d \in \mathcal{G}\) with \(d(x,p) = \infty\) or \(d(p,x) = \infty\).
(iii)
\(x \not = p\) implies \(\partial(x, \{p\}) = \infty\) or \(\partial(p, \{x\}) = \infty\).
This claim is false as (iii) does not imply (ii). This can be seen by the following example. Let \(X=\{x,p\}\) with \(x \not = p\). Let \(\mathcal{H} = \{d_\alpha \mid \alpha \in \mathbb{R}\}\) for \(d_\alpha\) defined by \(d_\alpha(x,p)= d_\alpha(p,x) = \alpha\) and zero elsewhere. Consider the gauge generated by the basis \(\mathcal{H}\). Then all \(d\) in the gauge take values in \(\mathbb{R}\) but clearly \(\partial(x,\{p\}) = \partial(p,\{x\}) = \infty\).
The same counterexample can be used to see that also Theorem 3.10. is false.
In Theorem 4.6 it is claimed that for an approach space \(X\) with gauge \(\mathcal{G}\) and its related distance \(\partial\), conditions (i) and (ii) are equivalent.
(i)
Every morphism from \(X\) to any discrete object is constant.
(ii)
For \(x \not = y\) in \(X\) there exists some \(d \in \mathcal{G}\) such that \(d(x,y) = d(y,x) = 0\).
The equivalence is false. Remark that condition (ii) holds in any approach space as the constant function, \(d = 0\) everywhere, always belongs to the gauge. Condition (i) does not always hold as we can see from the following example. Let \(X\) be a set with cardinality at least two and endow it with the discrete gauge \(\mathcal{G}_{dis} = \{ d \mid d \leq d_{dis}\}\) with \(d_{dis}(x,y) = \infty\) whenever \(x \not = y\) and \(d_{dis}\) is zero elsewhere. Then the identity map \(id: X \to X\) is a morphism that is not constant.
In Theorem 4.7. it is claimed that for an approach space \(X\) with gauge \(\mathcal{G}\) and its related distance \(\partial\), conditions (ii) and (iii) are equivalent.
(ii)
For \(x \not = y\) in \(X\) there exists some \(d \in \mathcal{G}\) such that \(d(x,y) = d(y,x) = 0\).
(iii)
For \(x \not = y\) in \(X\) one has \(\partial(x,\{y\}) = \partial(y,\{x\}) = 0\).
The equivalence is false. As we remarked earlier condition (ii) holds in any approach space. On the other hand condition (iii) implies that all \(d \in \mathcal{G}\) are identically zero. There is only one approach space fulfilling this condition, namely the indiscrete one with gauge \(\mathcal{G} = \{ 0\}\).
MSC:
54A05 Topological spaces and generalizations (closure spaces, etc.)
54B30 Categorical methods in general topology
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54E70 Probabilistic metric spaces
18B35 Preorders, orders, domains and lattices (viewed as categories)
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