Real Analysis, Lecture 11: Compact Sets
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when someone struggles to build intuition for compactness and open covers, and want a clear classroom explanation to supplement a textbook.
Recommended by 3 people across 3 different months, 2013 to 2018.
What it is
A full-length lecture from Francis Su's Spring 2010 Real Analysis course at Harvey Mudd College, recorded in class. It introduces compact sets in metric spaces, covering open covers, finite subcovers, and relative openness, following Rudin's approach.
What people said
Recommends kahen · 2013-04 · Mathematics Stack Exchange · on “Does the union of open neighborhoods of all points in a metric space cover the metric space?”
In response to your latest edit: I see you can make open covers of $M$, but I see *nothing* in your question that hints at how you want to extract finite subcovers. Might I suggest you watch Professor Su's presentation of compactness on Youtube. Lectures 11 [this video] , 12 [LINK] and 13 [LINK] . If I recall correctly, he's teaching from "Baby Rudin".
Recommends The very fluffy Panda · 2014-04 · Mathematics Stack Exchange · on “Developing an intuition for compact and open sets”
In the beginning it is always hard to understand these concepts, since they seem to lack the motivation. Don't worry you'll get the hang of it. The best way to learn mathematics is by doing it. Don't just sit around and think about these concepts, try to apply them. Try and solve a problem on your own, doesn't matter if it takes weeks, that's the only way you'll ever learn. I found these videos on Youtube, which I think are quite useful. This one [this video] is particularly on Compact Sets. The whole lecture series [LINK] is amazing. I think he talks about open sets at around Lecture 8 or 9.
Recommends user170231 · 2015-09 · Mathematics Stack Exchange · on “What does it REALLY mean for a metric space to be compact?”
I found this [this video] video lecture very helpful as a supplement to my undergrad real analysis course in giving me a visual sense of compact sets.
Cites TheLast Cipher · 2018-07 · Mathematics Stack Exchange · on “Regarding relative open sets.”
I'm watching this lecture by Francis Su at HarveyMudd and I am having trouble understanding why the pink subset $P$ of $Y$ is not open relative to $X$. I think I understand that $P$ is open relative to $Y$ because "relative to" changes the universe of discourse to $Y$ from $X$ so every $p\in P$ has an $\epsilon$ neighborhood $N(p,\epsilon)$ that is contained in $Y$. However, the class later agreed that $P$ is not open relative to $X$. How could this be when everything in $Y$ is also in $X$? Link to lecture(@1:02:07): [this video] A similar picture he has drawn since video quality is terrible: [LINK]