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Another approach to connectedness with respect to a closure operator. (English) Zbl 1184.54014

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 (categorical) closure operator on \(\mathcal X\) if the following conditions are fulfilled for each \(\mathcal X\)-object \({X}\) and each \({m,p} \in\) sub\({X}\): 6mm
(1)
\(m\leq {c}_X(m)\),
(2)
\(m\leq {p} \Rightarrow \, {c}_{X}(m) \leq c_{X}(p)\),
(3)
\( f({c_X}(m))\leq {c_Y}(f(m))\) for every \(\mathcal X\)-morphism \(f: X \rightarrow Y\).
In this paper a new concept of 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.

MSC:

54B30 Categorical methods in general topology
18D35 Structured objects in a category (MSC2010)
54A05 Topological spaces and generalizations (closure spaces, etc.)
54D05 Connected and locally connected spaces (general aspects)
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