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

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(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

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