Another approach to connectedness with respect to a closure operator (Q843761)

In this paper it is assumed \(\mathcal X\) is a finetely complete category with a proper (\(\mathcal E, \mathcal M)\)-factorization structure for morphisms, where \(\mathcal E\) is a class of epimorphisms and \(\mathcal M\) is a class of monomorphisms in \(\mathcal X\). Further it is assumed that \(\mathcal E\) is stable under pullbacks along \(\mathcal M\)-morphisms. For each \(\mathcal X\)-object \({X}\) sub\,\({X}\) means the subobject semilattice of \({X}\) and it is supposed that each sub\,\({X}\) has a least element. A family of maps \({c}\) = \((c{_X}\): sub\({X} \rightarrow\) sub\({X}) ( X\in\mathcal {X}\)) is said to be a \textit{(categorical) closure operator on} \(\mathcal X\) if the following conditions are fulfilled for each \(\mathcal X\)-object \({X}\) and each \({m,p} \in\) sub\({X}\): {\parindent6mm \begin{itemize}\item[(1)] \(m\leq {c}_X(m)\), \item[(2)] \(m\leq {p} \Rightarrow \, {c}_{X}(m) \leq c_{X}(p)\), \item[(3)] \( f({c_X}(m))\leq {c_Y}(f(m))\) for every \(\mathcal X\)-morphism \(f: X \rightarrow Y\). \end{itemize}} In this paper a new concept of \textit{connectedness} with respect to a categorical closure operator is introduced and the main result is a presentation of the theorem that this conectedness is preserved, provided that some natural conditions are fulfilled, by inverse images of subobjects under quotient morphisms. In addition an application of this result in digital topology is discussed.