What are Metric Spaces and Limit Points?

This video introduces metric spaces, open sets, limit points and closed sets.

//Books

//Exercises
- Prove the product to sum identities for exponentials first for rational numbers then for real numbers.

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0:00 Start
1:35 Neighborhoods and Open Sets
2:39 Limit Points and Closed Sets
4:17 Metric Spaces and Topologies
4:43 A Mistake in Baby Rudin
9:16 Wrap Up

[adokwuondoma]

Wow... great content Sir. Though I felt lost from the beginning until I had to watch at half the original speed and pauses in between. Thanks for sharing.

[jackbrolin]

Production is getting better and better! These awesome videos shine much needed light on topics only lightly touched in my analysis class.

[anerudhan]

Did you flip your camera around or did you just start wearing your watch on your right hand instead of left?

[dans]

isn't the concept of open and closed sets, as you described it, closely analogous to open and closed intervals on the Reals?

[mattlydon]

I get super excited when I see you have a new video out. I'm working through some analysis courses this term, and your videos are consistently fantastic resources.

[rikin]

This is such good timing considering I studied this just yesterday! Great video 🎉

[hansisbrucker]

I learned so much today.

Btw, can I say that I *love* that Spider-Man poster behind you 😁

[Cyclonus-fcxx]

Great video, as always, I'm not anywhere near having finished working through Rudin, but for the Topology part of Analysis I definitely prefer Pugh's Real Mathematical Analysis treatment of it, as he uses visual representations much more often, which really helps with something like Topology otherwise it gets way too abstract

[yingzhaolian]

Nice video. It would be great if you could talk a bit about the classification of the topological space. I find it hard to get a good intuition about how their difference can impact the proof or convergence.

[dasdawfda]

Will be doing a real analysis course this semester and I'm already saving all this videos, I believe they will be quite useful. Thanks!
Also, we'll be using understanding analysis by abbot, do you know it?

[darkpikachu_.]

cool, i just started reading topology from munkres, thanks for the content

[okoyoso]

Is there standard notation for the set of limit points? I've been using lim(E)