Today we’ll talk about how one “closes” an open set. Intuitively, this should be possible by just “adding” the open set’s boundary to the set. This brings up new concepts and relationships we’ll discuss today.
In Module I we’ll define the set of adherent points to a set S, and later see that we “close” S by adding in all of its adherent points. This will lead us to the notion of the closure of a set. Then, in Module II we’ll define what we mean by the boundary of a set, and relate that to the closure of the set. Finally, in Module III we’ll give another useful characterization of what it means to “close” a set by studying how points inside the set “accumulate.”
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.
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?
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.