The notions of closedness and \(D\)-connectedness in quantale-valued approach spaces.

*(English)*Zbl 1450.54002Approach 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.

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.

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.

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 \(\partial(p, \{x\}) = \infty\).

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 \(d(x,y) = d(y,x) = 0\).

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 \(\partial(x,\{y\}) = \partial(y,\{x\}) = 0\).

Reviewer: Eva Colebunders (Brussel)

##### 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) |

##### Keywords:

\(\mathcal{L}\)-approach distance space; \(\mathcal{L}\)-gauge space; topological category; separation; closedness; \(D\)-connectedness
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\textit{M. Qasim} and \textit{S. Özkan}, Categ. Gen. Algebr. Struct. Appl. 12, No. 1, 149--173 (2020; Zbl 1450.54002)

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