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
\(m\leq {c}_X(m)\),
\(m\leq {p} \Rightarrow \, {c}_{X}(m) \leq c_{X}(p)\),
\( f({c_X}(m))\leq {c_Y}(f(m))\) 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.


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)
Full Text: DOI


[1] Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. Wiley, New York (1990) · Zbl 0695.18001
[2] Castellini, G.: Connectedness with respect to a closure operator. Appl. Categ. Structures 9, 285–302 (2001) · Zbl 0981.18009
[3] Castellini, G.: Categorical Closure Operators. Birkhäuser, Boston (2003) · Zbl 1045.18001
[4] Castellini, G., Hajek, D.: Closure operators and connectedness. Topology Appl. 55, 29–45 (1994) · Zbl 0791.54001
[5] Castellini, G., Holgate, D.: A link between two connectedness notions. Appl. Categ. Structures 11, 473–486 (2003) · Zbl 1039.18001
[6] Clementino, M.M., Giuli, E., Tholen, W.: Topology in a category: compactness. Portugal Math. 53, 397–433 (1996) · Zbl 0877.18002
[7] Clementino, M.M., Tholen, W.: Separation versus connectedness. Topol. its Appl. 75, 143–181 (1997) · Zbl 0906.18003
[8] Clementino, M.M., Giuli, E., Tholen, W.: What is a quotient map with respect to a closure operator. Appl. Categ. Structures 9, 139–151 (2001) · Zbl 0978.18003
[9] Clementino, M.M.: On connectedness via closure operator. Appl. Categ. Structures 9, 539–556 (2001) · Zbl 0993.18004
[10] Dikranjan, D., Giuli, E.: Closure operators I. Topology Appl. 7, 129–143 (1987) · Zbl 0634.54008
[11] Dikranjan, D., Giuli, E., Tholen, W.: Closure operators II. In: Proceedings of the Conference Categorical Topology and its Relations to Analysis, Algebra and Topology, Prague 1988, pp. 297–335. World Scientific, Teaneck (1989)
[12] Dikranjan, D., Tholen, W.: Categorical Structure of Closure Operators. Kluwer Academic, Dordrecht (1995) · Zbl 0853.18002
[13] Engelking, R.: General Topology. Heldermann, Berlin (2003)
[14] Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (2003) · Zbl 1152.06300
[15] Giuli, E., Tholen, W.: Openness with respect to a closure operator. Appl. Categ. Structures 8, 487–502 (2000) · Zbl 0981.18001
[16] Giuli, E.: On m-separated projection spaces. Appl. Categ. Structures 2, 91–100 (1994) · Zbl 0840.18003
[17] Giuli, E., Šlapal, J.: Raster convergence with respect to a closure operator. Cahiers Topologie Géom. Différentielle Catég. 46, 275–300 (2006)
[18] Janelidze, G., Tholen, W.: Facets of descents I. Appl. Categ. Structures 2, 245–281 (1994) · Zbl 0805.18005
[19] Khalimsky, E.D., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topology Appl. 36, 1–17 (1990) · Zbl 0709.54017
[20] Preuss, G.: Trennung und Zusammenhang. Monatsh. Math. 74, 70–87 (1970) · Zbl 0185.26003
[21] Preuss, G.: Eine Galois-Korrespondenz in der Topologie. Monatsh. Math. 75, 447–452 (1971) · Zbl 0231.54014
[22] Preuss, G.: Relative connectedness and disconnectedness in topological categories. Quaestiones Math. 2, 297–306 (1977)
[23] Preuss, G.: Connection properties in topological categories and related topics. In: Proceedings of the Conference Categorical Topology, Berlin 1978. Lecture Notes in Mathematics 719, pp. 293–305. Springer, Berlin (1979)
[24] Šlapal, J.: A digital analogue of the Jordan curve theorem. Discrete Appl. Math. 139, 231–251 (2004) · Zbl 1062.54001
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