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The covering property axiom CPA is consistent with ZFC: it is satisfied in the iterated perfect set model. We show that CPA implies that for every ν∈ω∪{∞} there exists a family Fν⊂Cν(R) of cardinality ω1

Based on the Zermelo-Fraenkel system of axioms ZF, we introduce a theory of essential closures. It is a generalization of the concept of topological closures in which a set may not be contained in its essential closure. A typical essential closure collects all points that are essential with respect to a submeasure; hence it is called a submeasure closure. One of our main results states that a "nice" essential closure must be a submeasure closure. Many examples of known and new submeasure closures are discussed and their applications are demonstrated, especially in the study of the supports of measures. [ABSTRACT FROM AUTHOR]