Intro to Open Sets (with Examples) | Real Analysis

We introduce open sets in the context of the real numbers, along with examples and nonexamples of open sets. This is an important topic in the topology of the reals. We say a subset U of the reals is open if, for any x in U, there exists a delta-neighborhood of x that is contained in U. We'll prove that the real numbers are open, the empty set is open, and more! #realanalysis

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[isobaric]

My first introduction to open sets was in my metric spaces course, this video definitely helped simplify the concept for me. Thank you for the great video.

[jonathanabraham]

Thank you doctor you save me at this point before final

[zhengyangfei]

I just wanna appreciate your mic dude.

[aydenzhu]

Could you make a video for the following question: For part (a), show that f(x)=|x| is not differentiable at x=0. For part (b), show that if f: R–>R is differentiable at x0, then f is continuous at x0. Tank you very much!

[jayasuryav]

can you explain in detail about the null set is an open set

[cheyennehu]

this man talks math like an asmrist lol but i love it

[Nuuker]

I have a question: to say that the set X = [0, 1] it's not open, we have to say that X is a subset of another set, such as R for exemple? Because, if we think that [0, 1] is the entire space ("universe" space) when we make a open ball in point {1} for exemple there's no other space such that a point not belongs to [0, 1], in this case, a open ball will contain points that only belongs to X.
Is this correct or there's some error in this argument?

[Dravignor]

If { } is open, does that imply the universal set is closed?

[samtux]

Formaly. Here we limit ourselves to open sets. A number of theorems don't apply to closed sets like [0, 1] or semi-open sets like [0, 1). Open sets must have some nice properties. Nice enough, that we study them separately. After all, we don't prove theorems for sets (3, 5] (those are not general enough).

Tl;dr "What motivates mathematicians to impose such an strange requirement on sets? After all, we just exclude two points (points {0} and {1} from a continuum set of points)?"

[punditgi]

Let's get real with Wrath of Math! 😀