Way Below and Way Above Relation for Quotient Objects of Compacta
DOI:
https://doi.org/10.31471/2304-7399-2026-22(83)-88-94Keywords:
``Way below'' relation, ``way above'' relation, approximation relations, quotient object, compactumAbstract
The paper investigates the approximation relations “way below” and “way above” in the lattice of quotient objects of a compact Hausdorff space. Quotient objects are treated as equivalence classes of continuous surjections. The study focuses on structural links between these relations and properties of the corresponding factor maps. Necessary and sufficient conditions are obtained for one quotient object to approximate another from below. Exactly, we prove necessary and sufficient conditions for a quotient object of a compact Hausdorff space to be ``way below'' or ``way above'' another its quotient object. It is shown that finiteness of the image plays a crucial role in this case. For the “way above” relation, a characterization is given in terms of equivalence relations with finite support and isolated points. The results contribute to a deeper understanding of approximation relations in the theory of partially ordered sets and topological algebra.References
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Copyright (c) 2026 Kateryna Koporkh, Oksana Mykytsey
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