The idea is that by that procedure, every choice of open ball corresponding to an interior point by definition is contained in the open set. In the example, as the open balls are picked so that they are always contained within the triangle, they cover the triangle exactly. Hope this makes it a bit clearer

It's all made on Manim in Python!

I agree 100%, would like to add that itˋs a little confusing that the terms *don´t* actually move around as the animation just morphs it completely, would be a lot clearer if the terms actually moved around

@@ThomasEdits Yeah, I noticed that after writing my comment but didn't bother to edit it

By the way, the music is perfect

Ah I'm sorry to hear that. For me, it was that mathematicians abstracted distance by creating the definition of a metric space with the distance function relying on 4 key properties outlined in the video. Those 4 properties are what abstract distance, but I apologize if the video was in anyway misleading.

@@pi_squared2 Don't stress over it, worst case scenario one could call it "clickbait", which honestly isn't that bad nowadays anyway. As long as the content is there, it's mostly okay!

Thank you so much!

Lol

😂

Glad you enjoyed it!

Thank you, and yes that's a good correction! I've put it in the description of the video.

Sorry if it felt misleading, the abstraction of distance are the 4 properties listed under the distance function of a metric space. These 4 properties allow you to construct various different possible distance functions, some of them quite different to how we ordinarily measure distance in daily life.

also he should have included the property 1 as follows: d(x,y) = 0 <=> x=y Otherwise we have no metric but a pseudo metric

@@rielblakcori971 I believe that was encoded as d(x,x)=0.

@@mathunt1130 those are not the same since you can only imply right hand side to left hand side by his definition but no equivalence

@@rielblakcori971 As I showed, from d(x,x)=0, d(x,y)=d(y,x), and the triangle ineqality, I demonstrated that d(x,y)>=0. From that, we get your condition.

@@mathunt1130 yes but you still didn't show why you get from d(x,y)=0 that this implies x=y, that's the whole point

2 elements a and b of the set A are at distance 1 if the distance function outputs the real number 1 when aplied to a and b. There is no single function of distance, you can use any function from A to the real numbers that verifies those properties. The set A can be as weird as you want it to be, and different distance functions applied to a and b can output different values.

@@Xmodg4m3 Agree. There are infinitely many distance functions but the point I'm making is that author *escapes* from bringing any real example of such a function. We can pick 2 arbitrary points A, B on a plain and declare distance of 1 unit. But what property constites distance of 2 between B and C? The qq is how to build such a function.

@@kcg26876 A nice thing about math is that most of the time you are not working with concrete things, but with abstractions and generalizations instead. This is a neat thing as it replaces calculation with thinking. You are now for example proving things not just about the real space R^3, but about any abstract vector space. This makes so that when you are working with a really abstract and complicated object, but that can be seen as a vector space, you automatically unlock all the vast knowledge humanity has about vector spaces and can apply it right away, whithout needing years of research just for your example. He is not fleeing the need to define a concrete distance, as there is no such need to do so. It's just working the way up from a general concept to reach truths that can be applied whenever you try to solve any kind of real life problem involving any kind of notion of distance.