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Based on conditional set theory, we study conditional weak topologies,extending some well-known results to this framework and culminating with theproof of conditional versions of Eberlein-\v{S}mulian and Amir-LindenstraussTheorems. In pursuing this aim, we prove conditional versions of Baire CategoryTheorem and Uniform Boundedness Principle.

Versions of Eberlein-\v{S}mulian and Amir-Lindenstrauss theorems in the framework of conditional sets