American Institute of Mathematical Sciences

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March  2020, 28(1): 471-548. doi: 10.3934/era.2020027

Generalizations of some ordinary and extreme connectedness properties of topological spaces to relator spaces

Muwafaq Salih , and  Árpád Száz

Institute of Mathematics, University of Debrecen, H-4032 Debrecen, Egyetem tér 1, Hungary

* Corresponding author: Muwafaq Salih

Dedicated to the memory of János Kurdics
who was the first to note that connectedness is a particular case of well-chainedness

Received  December 2019 Revised  February 2020 Published  March 2020

Fund Project: The work of the second author has been partially supported by the Hungarian Scientific Research Fund (OTKA) Grant K-111651

Motivated by some ordinary and extreme connectedness properties of topologies, we introduce several reasonable connectedness properties of relators (families of relations). Moreover, we establish some intimate connections among these properties.

More concretely, we investigate relationships among various minimalness (well-chainedness), connectedness, hyper- and ultra-connectedness, door, superset, submaximality and resolvability properties of relators.

Since most generalized topologies and all proper stacks (ascending systems) can be derived from preorder relators, the results obtained greatly extends some standard results on topologies. Moreover, they are also closely related to some former results on well-chained and connected uniformities.

Keywords: Generalized uniformities, well-chained, connected, hyperconnected, ultraconnected, submaximal and resolvable spaces.
Mathematics Subject Classification: Primary: 54E15, 54D05; Secondary: 54G15, 54G20.
Citation: Muwafaq Salih, Árpád Száz. Generalizations of some ordinary and extreme connectedness properties of topological spaces to relator spaces. Electronic Research Archive, 2020, 28 (1) : 471-548. doi: 10.3934/era.2020027
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