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The co-elementary hierarchy is a nested ordinal-indexed sequence of classes of mappings between compacta, with each successor level being defined inductively from the previous one using the topological ultracopower construction. The lowest level is the class of continuous surjections; and the next level up, the coexistential maps, is already a much more restricted class. Co-existential maps are weakly confluent, and monotone when their images are locally connected. These maps also preserve important topological properties, such as: being infinite, being of covering dimensions ≤ n, and being a (hereditarily decomposable, indecomposable, hereditarily indecomposable) continuum.
Bankston, Paul, "Continua and the Co-elementary Hierarchy of Maps" (2000). Mathematics, Statistics and Computer Science Faculty Research and Publications. 185.