So Hausdorff criteria makes calculus possible as well defined, unique limits. Brilliant lecture, thank you! The examples chosen for this series are highly instructive as is the guidance throughout! This is the standard university pedagogy should be at.
As the feeble joke goes, a Hausdorff space is one where different points can be "housed off" into different open sets. (best read in a plummy English accent: "housed orf") If we're only at Hausdorff spaces, it looks like this is going to be an impressively long series by the time we get to tensors. Looks very promising so far though.
You are right. It will take time to tackle all the problems in the series. However, I don't want to rush through the things. Therefore, I gladly invest a lot of videos here :)
@@brightsideofmaths I'm amazed at how quickly you produce videos of such quality. I am making videos on far more elementary topics, and I can't replicate your rate of work. Do you teach as well?
Hi , thanks for this video. If I have correctly understood the example, I could say that it does not matter which non-positive real number I can choose as a limit of the sequence (you call it "a" ) , because there will be always a negative real number "b" such that, the set U=(b,inf), belonging to the topology, will include the sequence for any N (as big as I want).
I remember reading somewhere a while ago that in certain spaces, there are sequences which can converge to multiple points simultaneously, and I was so confused at the time... Now I know what it was about :)
hey! in the topological space of the example, isnt there suposed for all intersections to be in the collection? Is that for example the intersection of (a,infinity) and (b,infinity), if a is diferent from b, is at least a half open half closed interval. Please do correct me if I'm mistaken.
@@brightsideofmaths oh god D: i was substracting the sets! 😅 oh well i'? Very glad i was wrong. Hey bright side, now that i've got your atention im gonna ask you a cuestion from another topic: can we talk about a set of the different existing cardinals? If so would it be countable or uncountable?
I can't grasp why in the example 5:20, the second set can not just be {(b, a) | b,a € R}. What is the difference with the infinity? The set (b, infinity) is still not equal to R and not in the topology T.
@@brightsideofmaths Why does the open set (with the exception of the empty set) always have to stretch to the infinity? If we have said: {ø, R} U {(b,a)} where both b and a are real numbers, isn't it also an empty set? Thanks for the answer!
Don't forget that Hausdorff spaces are still way more general than metric spaces. However, the general idea is that we disregard everything we don't need. Abstraction can make things easier :)
Hausdorff spaces? More like "The concepts in these videos put me through my paces!" Some of them seem to melt my brain, but if I spend enough time and understand them, I feel like I'm walking away with something really cool...
So Hausdorff criteria makes calculus possible as well defined, unique limits. Brilliant lecture, thank you! The examples chosen for this series are highly instructive as is the guidance throughout! This is the standard university pedagogy should be at.
Now the definition of Hausdorff Spaces really make sense. Thank you so much for making this video!!
As the feeble joke goes, a Hausdorff space is one where different points can be "housed off" into different open sets. (best read in a plummy English accent: "housed orf")
If we're only at Hausdorff spaces, it looks like this is going to be an impressively long series by the time we get to tensors. Looks very promising so far though.
You are right. It will take time to tackle all the problems in the series. However, I don't want to rush through the things. Therefore, I gladly invest a lot of videos here :)
@@brightsideofmaths I'm amazed at how quickly you produce videos of such quality. I am making videos on far more elementary topics, and I can't replicate your rate of work. Do you teach as well?
hausdorff spaces are my 2nd fav kind of topological space, and of course my first fav kind of space is a manifold :D
just wanted to express my happiness for understanding this more intuitively. I hope you understand how grateful we all are to you!
Thanks a lot :)
00:00 Intro
0:32 Convergence in metric and topological space
5:26 Example. Multiple limits of a sequence
7:39 Hausdorff space definition
Hi , thanks for this video. If I have correctly understood the example, I could say that it does not matter which non-positive real number I can choose as a limit of the sequence (you call it "a" ) , because there will be always a negative real number "b" such that, the set U=(b,inf), belonging to the topology, will include the sequence for any N (as big as I want).
Almost! The point is that you cannot find any open neighbourhood of a such that the sequence lies outside it.
@@brightsideofmaths thanks, for the further explanation.
You are such a great teacher, thanks man!
this is so weird: a limit can be non-unique. it makes sense given the example but its very strange. love it!
In the trivial topology, any sequence will converge to all points :)
Such a clear presentation, very well done!
Many times, the non-examples you give are much more revealing than the actual examples.
I'm really enjoying this series
I remember reading somewhere a while ago that in certain spaces, there are sequences which can converge to multiple points simultaneously, and I was so confused at the time... Now I know what it was about :)
In the trivial topology, any sequence will converge to all points :)
Love this channel. Everything is explained so simply. Thanks for your videos and the amazing work you do. Kind regards from Ásgeir in Iceland.
Thank you very much :)
this my first time watching your video. your explian is very simplefied and nice 👍
Thank you so much 🙂
Wow I have been ever seen first that there is a multiple limit of a sequence in a topological space
Good video, very clear!
Thanks!
will you cover integration on manifolds?
Yes, of course :)
Thank you so much sir
Most welcome
Great ! Any non positive real numbers are a limit of the sequence
hey! in the topological space of the example, isnt there suposed for all intersections to be in the collection? Is that for example the intersection of (a,infinity) and (b,infinity), if a is diferent from b, is at least a half open half closed interval. Please do correct me if I'm mistaken.
(a,infinity) intersected with (a+1,infinity) is (a+1,infinity). Does this help you?
@@brightsideofmaths oh god D: i was substracting the sets! 😅 oh well i'? Very glad i was wrong. Hey bright side, now that i've got your atention im gonna ask you a cuestion from another topic: can we talk about a set of the different existing cardinals? If so would it be countable or uncountable?
So this is where the limit in multivariable calculus stops being sufficient?
I wouldn't say that because in multivariable calculus you should deal with Hausdorff spaces.
just out of curiosity, how long until you plan on getting to homeomorphisms?
Part 7, I think :)
I can't grasp why in the example 5:20, the second set can not just be {(b, a) | b,a € R}. What is the difference with the infinity? The set (b, infinity) is still not equal to R and not in the topology T.
Thanks for the question. I don't get it completely. What is the problem with the example?
@@brightsideofmaths Why does the open set (with the exception of the empty set) always have to stretch to the infinity? If we have said: {ø, R} U {(b,a)} where both b and a are real numbers, isn't it also an empty set?
Thanks for the answer!
Metric spaces aren't general enough! Let's invent topological spaces!
4 videos later:
Topological spaces are too general! Let's invent Hausdorff spaces!
Don't forget that Hausdorff spaces are still way more general than metric spaces.
However, the general idea is that we disregard everything we don't need. Abstraction can make things easier :)
I’ve watched the example many times and I simply do not understand how can it converge to a negative number.
Maybe it helps to draw the open sets mentioned here. A sketch really tells you something.
Union is dual to intersection.
Somebody know the board app which he use to do these videos?... 🤔
I know!
@@brightsideofmaths 😂😂😂 Do not ever change!
@@Juanchoxd0316LVJXD xournal i think
@@mastershooter64 thanks!
Buen video.
Hausdorff spaces? More like "The concepts in these videos put me through my paces!" Some of them seem to melt my brain, but if I spend enough time and understand them, I feel like I'm walking away with something really cool...
Hausdorff > Hofstadter