Splitting closure operators
The authors extend Arkhangel'skiĭ's notion of splitting space to complete \((\mathcal{E, M})\)-categories: Suppose that \(\mathcal{X}\) is such a category and \(\mathcal{P}\) is a subclass of \(\mathcal{X}\). An object \(X\) belongs to the splitting class of \(\mathcal{P}\) if and only if for every subobject \(m\) of \(X\) there exist an object \(P\in\mathcal{P}\) and a morphism \(f:X\to P\) such that \(m=f^{-1}(f(m))\) (i.e., every subobject is a \(\mathcal{P}\)-piece). In section 2 of the paper basic properties of splitting classes are presented. In Section~3 an idempotent closure operator \(\text{spl}^{\mathcal{P}}\) is constructed. Then a splitting closure operator is defined as a closure operator \(\mathbf{c}\) for which there exists a class \(\mathcal{P}\) such that \(\mathbf{c}= \text{spl}^{\mathcal{P}}\). The authors illustrate every notion introduced in this paper with examples in classical categories as the category of left modules, the category of topological spaces and the category of Tychonoff spaces.
scientific article; zbMATH DE number 5001793