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Real Analysis by H. L. Royden

524: Measure theory and Lebesgue integration
525: Topology and functional analysis
526: Measure theory and topology

An undergraduate course on real analysis, including coverage of metric spaces and some kind of integration theory, preferably Lebesgue integration.  Also, linear algebra and a little modern algebra, including what groups, rings, and ideals are.  (The latter can be taken concurrently; it is needed starting Winter quarter.)