Real Analysis Final Exam Review Problems and Solutions (Topology on Metric Spaces) скачать в хорошем качестве

Real Analysis Final Exam Review Problems and Solutions (Topology on Metric Spaces)

Definitions in a metric space (X,d): interior point, open set, limit point, closed set, open cover, finite subcover, compact set. Theorems: Heine-Borel Theorem, continuity in terms of preimages of open sets, image of a compact set is compact (Extreme Value Theorem). Examples: interior, closure, open sets, closed sets, compact sets. Proofs: 1) Triangle Inequality for sup norm (infinity norm), 2) Open balls are open sets, 3) espilon/delta definition of continuity implies preimages of open sets are open, 4) under a continuous map, the image of a compact set is compact (Extreme Value Theorem). https://amzn.to/3GgFjcc ("Real Analysis", by Russell Gordon) 🔴 Real Analysis Course Playlist:    • Introduction to Real Analysis Course Lectures   🔴 Abstract Algebra Course Playlist:    • Abstract (Modern) Algebra Course Lectures   🔴 Complex Analysis Course Playlist:    • Introduction to Complex Analysis Course Le...   #RealAnalysis #RealAnalysisFinal #RealAnalysisFinalExam Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter:   / billkinneymath   🔴 Follow me on Instagram:   / billkinneymath   🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ (0:00) Introduction (1:20) Interior point definition (in a metric space) (4:16) Open set definition (metric space) (7:00) Limit point definition (metric space) (11:52) Closed set definition (metric space) (17:52) Open cover of E definition (20:08) Finite subcover definition (or an open cover) (21:26) Compact set definition (every open cover has a finite subcover) (22:24) Heine-Borel Theorem (25:32) Preimage of an open set under a continuous map (32:09) Continuous image of a compact set is compact (continuity preserves compactness, generalizes the Extreme Value Theorem) (34:22) Examples of interiors, closures, open sets, closed sets, and compact sets (and non-examples) (45:16) Prove Triangle Inequality for the sup norm (infinity norm) on a function space (52:42) Prove an open ball is an open set (59:34) Prove continuous preimage of an open set is an open set (preimages are also called inverse images) (1:07:43) Prove continuous image of a compact set is compact AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.