Intro Real Analysis, Lec 33, Euclidean Metric, Triangle Inequality, Metric Spaces, Compact Sets

Topology Unit, Part 3. (0:00) Goals for the lecture. (0:28) Euclidean n-dimensional space. (5:00) Properties of metrics are satisfied by the Euclidean metric. (6:58) Sketch the proof of the triangle inequality for the Euclidean norm and then the Euclidean metric. (17:03) Defining a topology (define open and closed sets) in a metric space. (27:39) A reminder to make sure you think beyond the standard pictorial examples. What is the interior of the rationals? What is the derived set of the rationals? (29:07) Continue with the definition of a closed set. (30:59) The empty set is both open and closed (make the truth table for an implication). (33:52) The closure of a set. (34:49) Any open ball is an open set (and give a sketch of the proof, which requires the triangle inequality). (44:52) Open covers, finite subcovers, compact sets. (52:13) In Euclidean space, a set being compact is equivalent to being closed and bounded (Heine-Borel Theorem). (55:38) Why are compact sets important? Discuss this in terms of the Extreme Value Theorem.