Manifolds 7 | Continuity
Вставка
- Опубліковано 2 лип 2024
- 📝 Find more here: tbsom.de/s/mf
👍 Support the channel on Steady: steadyhq.com/en/brightsideofm...
Other possibilities here: tbsom.de/sp
You can also support me via PayPal: paypal.me/brightmaths (Use "For friends and family" as payment type)
Or via Ko-fi: ko-fi.com/thebrightsideofmath...
Or via Patreon: / bsom
Or via other methods: thebrightsideofmathematics.co...
Please consider to support me if this video was helpful such that I can continue to produce them :)
Each supporter gets access to the additional material. If you need more information, just send me an email: tbsom.de/s/mail
Watch the whole video series about Manifolds and download PDF versions and quizzes: tbsom.de/s/mf
Supporting me via Steady is the best option for me and you. Please consider choosing a supporter package here: tbsom.de/s/subscribe
🌙 There is also a dark mode version of this video: • Manifolds 7 | Continui...
🔆 There is also a bright mode version of this video: • Manifolds 7 | Continuity
🔆 To find the UA-cam-Playlist, click here for the bright version: • Manifolds
🌙 And click here for the dark version of the playlist: • Manifolds [dark]
🙏 Thanks to all supporters! They are mentioned in the credits of the video :)
This is my video series about Manifolds where we start with topology, talk about differential forms and integration on manifolds, and end with the famous Stoke's theorem. I hope that it will help everyone who wants to learn about it.
x
#Manifolds
#Mathematics
#LearnMath
#calculus
I hope that this helps students, pupils and others. Have fun!
(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)
![Functional Analysis 12 | Continuity [dark version]](http://i.ytimg.com/vi/qr7sd3V-MoQ/mqdefault.jpg)
Having properly understand topology and seeing the definition of continuity, it is definitely a beautiful description to talk about the continuity imo.
I cannot thank you enough for the huge help in understanding these topics. I never got to fully understand them in class and now I'm getting the notion again of the beauty behind it.
You are so welcome!
There is an equivalent definition of topological continuity (preimages of open sets are open sets) which is a generalisation of the epsilon-delta-continuity:
For all neighbourhoods V of f(p) there is a neighbourhood U of p such that f(U) is a subset of V.
Thanks for your great videos as always !
Wow, the very high quality of these videos has some great...continuity!
😎
I apologize for not providing the timecodes as I've promised before, but the regime in my country just started a war against a sovereign nation and there're lots of problems (losing jobs, rising prices, other shortages) which effectively prevent me from thinking about math for some time. I'll try to contribute something occassionally, but it's highly probable that the internet access soon will be blocked.
Thanks for sharing and I wish you all the best.
Thanks Waiting for bundles
Hello and thank you for what you are doing. Question: why in your opinion open (and not closed, for example) sets are so important for continuity and for topology? Don't the near elements can also be found using closed balls?
Love your videos! BTW What kind of software and hardware do you use to do the note taking and graphics for the videos?
Xournal and a graphics tablet :)
Why are the only possible preimages of the indiscrete topology the empty and the whole set?
These are the only open sets there.
@@brightsideofmaths But only in Y, X can have other open sets right? So how do I know that an open set in X maps to the whole space and empty set in Y? If f is a bijection I understand that f(X)->Y but what about the empty set in Y?
@@chronos5271 Check again what continuous means: it's about the preimages and not the images.
@@brightsideofmaths Hmm I think I understand it now. Thank you for your help!
@@chronos5271 Maybe what you missed: the preimage of the empty set is always the empty set, and the preimage of the whole codomain is always the whole domain.
thanks for the great video. One question though, could you please elaborate on the definition of a continuous function. One might ask why it is not defined in terms of image instead of preimage, i.e., the image of an open set is an open set ?
Thanks! I think that the epsilon-delta-continuity I explain in my Real Analysis could help you. This here is just the abstraction of this idea.
The fact that the image of an open is an open set it's not a satisfactory way of characterizing continuous functions. Think for example in a constant function (seeing as a function from R to R with the usual topology). One surely want that this function would be continuous, however the image of an open set is a singleton that's not open.
This condition, in the other hand plays an important role. A function which verifies it it's called an open function. For example, as you can check a function is a homeomorphism iff is a bijection that is both continuous and open.
@@josemarino8787 thanks for the great reply and for the mention of open function. Indeed if we understand the continuous functions on topological spaces as a generalization of the usual continuous functions on the real line R, the definition makes perfect sense. But if we look at it from the structure preserving point of view, to me it’s a bit unintuitive, at least in comparison to linear maps from a vector space to another.
Hi. Thanks a lot.
Would you please explain why we needed to define topological spaces? Weren't metric spaces enough for mathematical needs?
I really appreciate if you answer this.
Yes, exactly! Metric spaces are sometimes too restrictive.
@@brightsideofmaths thanks. Then I think I should learn more to understand what topological spaces provide that metric spaces can't.
And I really appreciate your work. 🙏
謝謝！
Thanks :)
Is sequentially continuity aquivalent to continuity in Hausdorff-Spaces?
No, it's not equivalent.
@@brightsideofmaths Ok, thanks!
Useful stuff - but unless I'm dreaming it, didn't you cover this material in another playlist?
(And by the way, are you aware that a channel called "Y C" seems to have copied many of your videos? - measure theory, mainly, I think)
Yes, we also discussed continuity in my functional analysis series.
I am aware of the other channel. I will do something about it.
@@brightsideofmaths Right, I knew I'd seen it somewhere. I watch so many UA-cam maths videos, I tend to forget who did what. And I'm quite happy to see it all repeated - repetition is the key to learning tricky material.
Dear sir, at the time stamp 6:09 how was it guaranteed that f^-1(Y) is the whole set X itself? Since in general codomain has more elements than the domain. Please correct me!
This is the definition of a function or map.
@@brightsideofmaths Ok, so when we are considering f^-1 to exist here it means f is bijective. Hence my above argument holds true. Isn't it?
@@sayanjitb Don't confuse the _reciprocal image_ of a map, versus the _inverse image_ of a map.
The first one can _always_ be constructed from any map from set-alike structures to set-like structures. They are "sets"/collections of elements which are sent to the image collection of elements.
In contrast, an inverse map of a map f, will be uniquely defined if and only if there is a bijection between _for all_ image elements and pre-image elements. Namely, for each element in both the pre-image/co-domain and image/pre-image, I can construct both a map and its inverse. Note this applies for each element "locally".
Reciprocal maps are sending back the _whole packet,_ the whole image *set* to the original pre-image *set.* Not the elements one by one.
Inverse maps only deal with bijections _of elements_ which is too restrictive (in the case of topological spaces, one would to restrain to homeomorphisms).
However, reciprocal images are more general, and provide you the full set of pre-images. In other words, from the definition, it doesn't make sense to talk about "a" reciprocal image of an element, in the most general case.
@@sayanjitbbecause f(X) is part of Y and f-1 takes everything from Y which carries f(X) back to X
@@sayanjitbim not sure if the map has to be full
your video is very interesting, please can you give me an example of sequentially continuous but not continuous in topological spaces
Thanks. I don't get your question completely. Do you want topological spaces or convergent sequences inside one?
@@brightsideofmaths I want a sequentially continuous function but not continuous
@@khalidalisawi8037 Maybe here: math.stackexchange.com/questions/1412415/give-an-example-of-a-function-f-x-to-y-which-is-sequential-continuous-but-no
8:32 I am not 100% sure, but I think you might have forgotten to introduce the convergence (of sequence) concept in a topological space, i.e., without using metric.
It's in part 3 :)
@@brightsideofmaths Oh you are right. My bad. It would be helpful here to emphasize the convergence of the sequence is in a topological space.
@@shakesbeer00 Yeah, everything works in a topological space because no metric is defined. However, you might be right that it should be emphasized more often.
this series is great ... but I'm starting to think it should have been called "general topology"
True but I will take a turn soon :)
Part 9 will start with the definition of manifolds, finally :)
Yeah, it seems like that, but bear in mind that most differential geometry texts (for mathematicians, at least) start with this material, so it's the usual development. And of course, mathematicians study pure "topological manifolds", which are manifolds not equipped with any "differentiable" structure (so no vectors etc).
But you need the continuity (as defined by the underlying topology) else you can't define the differentiable stuff on top.
@@brightsideofmaths It would be great if you can cover connectedness as well before turning to Manifolds. I think Connectedness is another property that is kinda crucial
is (xₙ)n∈N ⊆ X proper notation? Since sequences are actually functions from the natural numbers to X, and functions are sets of tuples, sequences don't directly contain elements of X as its elements and therefore a sequence can't be a subset of X?
You are right, it's a little bit abuse of notation, but it is so common and a good abbreviation that one can definitely use it all the time.