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A generally agreed upon requirement for correctness of concurrent executions in Transactional Memory systems is that all transactions including the aborted ones read consistent values. Opacity is a recently proposed correctness criterion that satisfies the above requirement. Our first contribution in this paper is extending the opacity definition for closed nested transactions. Secondly, we define conflicts appropriate for optimistic executions which are commonly used in Software Transactional Memory systems. Using these conflicts, we define a restricted, conflict-preserving, class of opacity for closed nested transactions the membership of which can be tested in polynomial time. As our third contribution, we propose a correctness criterion that defines a class of schedules where aborted transactions do not affect consistency of the other transactions. We define a conflict-preserving subclass of this class as well. Both the class definitions and the conflict definition are new for nested transactions.