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Towards the theory of \(L\)-bornological spaces. (English) Zbl 1253.46077

The authors introduce the concept of an \(L\)-bornology on a set \(X\), where \(L\) is an infinitely distributive complete lattice, namely, a family \(\mathcal B\subseteq L^X\) such that
(1)
\(\bigvee \{B:B\in \mathcal B\}=1_x\);
(2)
\((B\in \mathcal B)\land (C\in L^X) \land (C\leq B)\Rightarrow C\in \mathcal B\);
(3)
\(B_1,B_2\in \mathcal B\Rightarrow B_1\vee B_2\in \mathcal B\).
Then the pair \((X,\mathcal B)\) is called an \(L\)-bornological space. Especially, when \(L\) is a two-point lattice, an \(L\)-bornological space is just a classical bornological space. The paper also proves that \(B(X,L)\), the family of all bornologies on \(X\), is a complete infinitely distributive lattice with respect to the partial order \(\preccurlyeq :\mathcal B_1 \preccurlyeq \mathcal B_2\Leftrightarrow \mathcal B_2\subseteq \mathcal B_1\) (\(\mathcal B_1,\mathcal B_2\in \mathcal B(X,L)\)); besides, the weakest \(L\)-bornology on \(X\) is \(L^X\) and the strongest \(L\)-bornology on \(X\) is BT (here I omit the concrete expression). The last part of this paper discusses Born\((L)\), that is, the category of \(L\)-bornological spaces, and obtains the final structures and initial structures in Born\((L)\) (Theorems 5.1 and 5.2).

MSC:

46S40 Fuzzy functional analysis
46A17 Bornologies and related structures; Mackey convergence, etc.

Keywords:

\(L\)-bornology
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