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Explore the third lecture in a comprehensive series on reverse mathematics for analysis, delivered as part of the Summer School on "Reverse Mathematics: New Paradigms." Begin by reviewing the "big five" subsystems of second-order arithmetic and their fundamental properties, then establish precise definitions of open and closed sets alongside continuous functions within Polish spaces. Delve into the reverse mathematical analysis of basic analytical concepts, examining how various mathematical theorems relate to different subsystems of second-order arithmetic. Investigate extended definitions of functions on Polish spaces to uncover additional analytical topics, gaining insight into how the strength of mathematical axioms corresponds to the complexity of analytical results in this specialized field of mathematical logic.
