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

Muwafaq Salih; Árpád Száz; Muwafaq Salih; Árpád Száz

doi:10.3934/era.2020027

Electronic Research Archive

2020, Volume 28, Issue 1: 471-548. doi: 10.3934/era.2020027

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

Muwafaq Salih ^,,
Árpád Száz

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

Received: 01 December 2019 Revised: 01 February 2020
Primary: 54E15, 54D05; Secondary: 54G15, 54G20

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.
- Generalized uniformities,
- well-chained,
- connected,
- hyperconnected,
- ultraconnected,
- submaximal and resolvable spaces
Citation: Muwafaq Salih, Árpád Száz. Generalizations of some ordinary and extreme connectedness properties of topological spaces to relator spaces[J]. Electronic Research Archive, 2020, 28(1): 471-548. doi: 10.3934/era.2020027

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Abstract

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.

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