When a disconnected space is made up of several smaller connected spaces, then each smaller component is called a "connected component" of the larger space.
See for example the
There are two cases:
- (topological) manifolds
- differential manifolds
Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?
- Original problem posed, for topological manifolds.AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.
- for differential manifolds:Counter examples are called exotic spheres.Totally unpredictable count table:is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??
Video "Finite fields made easy by Randell Heyman (2015)" at youtu.be/z9bTzjy4SCg?t=159 shows how for order . Basically, for order , we take:For a worked out example, see: GF(4).
- each element is a polynomial in , , the polynomial ring over the finite field with degree smaller than . We've just seen how to construct for prime above, so we're good there.
- addition works element-wise modulo on
- multiplication is done modulo an irreducible polynomial of order
So simple!! You can either:
- cut two holes and glue a handle. This is easy to visualize as it can be embedded in : you just get a Torus, then a double torus, and so on
- cut a single hole and glue a Möbius strip in it. Keep in mind that this is possible because the Möbius strip has a single boundary just like the hole you just cut. This leads to another infinite family that starts with:
You can glue a Mobius strip into a single hole in dimension larger than 3! And it gives you a Klein bottle!
Intuitively speaking, they can be sees as the smooth surfaces in N-dimensional space (called an embedding), such that deforming them is allowed. 4-dimensions is enough to embed cover all the cases: 3 is not enough because of the Klein bottle and family.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact