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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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