σ-Continuity and related forcings.

ALGEBRA; BOREL sets; BOOLEAN algebra; BAIRE spaces; MATHEMATICAL functions

The Steprāns forcing notion arises as quotient of the algebra of Borel sets modulo the ideal of σ-continuity of a certain Borel not σ-continuous function. We give a characterization of this forcing in the language of trees and use this characterization to establish such properties of the forcing as fusion and continuous reading of names. Although the latter property is usually implied by the fact that the associated ideal is generated by closed sets, we show that it is not the case with Steprāns forcing. We also establish a connection between Steprāns forcing and Miller forcing thus giving a new description of the latter. Eventually, we exhibit a variety of forcing notions which do not have continuous reading of names in any presentation. [ABSTRACT FROM AUTHOR]

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