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Explore the fourth 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. Investigate how these mathematical foundations apply to reverse mathematics in basic analysis, examining the logical strength required for various analytical theorems. Discover how function definitions on Polish spaces can be extended to explore additional analytical topics, gaining insight into the intricate relationships between logical systems and mathematical analysis through this systematic approach to reverse mathematics.
