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Preview
Preview
We’ve now learned a lot about the topology of Rn, and in particular how to “close” a set. Some of the most important theorems in calculus assume we’re working inside a closed interval (e.g., the Extreme Value Theorem). Today we’ll being our study of what makes closed intervals—or, more generally, closed and bounded sets—so special.
We’ll first discuss open coverings of a set (Module I). Then we’ll prove the Lindelof Covering Theorem (Module II), which says that any open cover of a set contains a countable subcover. We’ll use that to later introduce compactness, a concept that’ll come back when we prove the Extreme Value Theorem.
Review
Review
Learn
Learn
Work through the lesson notes below. In class, I will fill in the "See Class Notes" boxes. Optional: Watch the video(s) below the embedded notes to get another instructor's take on the content covered in the notes.
Some tips for you as you work through these resources:
I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.
Lesson Notes
Lesson Notes
Lesson 12.pdfOptional Resources
Optional Resources
Stop at compactness definition
Reflect
Reflect
If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:
Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.
Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.
If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:
Please summarize the main mathematical takeaways from the lesson notes.
What was the most interesting part of what you learned, and why?
What, if anything, do you still find confusing?
Practice
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood. Book problems come from the book for the course, Mathematical Analysis, Second Edition, by Tom Apostol.
Lesson 12 PP.pdfPage updated
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