6:23: the 6th line on the paper which says "V closed open imply V=0 or U" seems very magical to me . Why define V to be the largest open ,and what used of it is not very clear to me . My understanding to the meaning of it is that maybe V is some subset closed inside and open on the boundary ,so maybe the boundary cannot inside U because of the holomorphic ? besides, U is close and V is open , how can U=V ? What if V has some funny fractal shape like boundary . I cannot picture it very well.
V is closed as a subset of U, and it is also open as a subset of U, in the sense of subset topology on U. This means that V contains all of its limit points on U( this is the definition of closed subset) and V is open because it's a union of open sets. Because U is connected, an alternative definition of connectedness is that the only subsets which are closed and open (as subsets of U) are U and the empty set. Since V is both closed and open and not empty, it necessarily has to be U. The "largest" only means that it's the union of all of the sets that have the property.
In the standard topology in C (same is true for R^n), the only sets that are both closed and open are φ and the whole space. In this case, as explained by Monny18, the whole space is the domain U. You can see this as follows: Let the space be X. It’s easy to check that φ and X are both closed and open. If U is a proper non-empty subset that is both closed and open, pick a point p that’s not in U and let x be the closest point to p in U, in the sense of length of path joining p (connectedness is used here). Then by closedness, x is a boundary point of U. However by openness of U there’s a neighborhood of x that lies inside U, which contradicts the choice of x since we can find another point that is closer to p in neighborhood of x. Note that closedness of U also guarantees that distance of p and x is positive, otherwise p will be a member of U. This result is not true in general topological space that is not connected, say X = {1,2} with discrete topology.
@@antonios-alexandrosrobotis8205 These lectures prompted me to dig deeper, and I've just learned, that they can be approximated nicely by tropical algebra and tropical geometry. On the other hand one is rather troubled by young fast and furious fitting big data onto them.
Please don't pollute a study space with comments like these. It's highly unprofessional. We should be thankful that Prof. Borcherds is spending so much of his time in the exposition of this material.
@@sjhd98 Apophis is just showing his excitement/appreciation for the content. This is a RUclips video, not a formal academic setting, and so some degree of informality is appropriate. I don't think there's any need to get all elitist about it.
@@ruinenlust_ It's harmless. Any perceived dearth of engaging comments is not due to useless comments 'polluting' the comment section of Borcherds' videos.
Every morning these lectures come out like news of the day!
6:23: the 6th line on the paper which says "V closed open imply V=0 or U" seems very magical to me . Why define V to be the largest open ,and what used of it is not very clear to me . My understanding to the meaning of it is that maybe V is some subset closed inside and open on the boundary ,so maybe the boundary cannot inside U because of the holomorphic ? besides, U is close and V is open , how can U=V ? What if V has some funny fractal shape like boundary . I cannot picture it very well.
Closed and open aren't mutually exclusive, you can find sets which are neither closed nor open or both. Try to picture them, it's a funny exercise 👍
V is closed as a subset of U, and it is also open as a subset of U, in the sense of subset topology on U. This means that V contains all of its limit points on U( this is the definition of closed subset) and V is open because it's a union of open sets. Because U is connected, an alternative definition of connectedness is that the only subsets which are closed and open (as subsets of U) are U and the empty set. Since V is both closed and open and not empty, it necessarily has to be U. The "largest" only means that it's the union of all of the sets that have the property.
In the standard topology in C (same is true for R^n), the only sets that are both closed and open are φ and the whole space. In this case, as explained by Monny18, the whole space is the domain U.
You can see this as follows: Let the space be X. It’s easy to check that φ and X are both closed and open. If U is a proper non-empty subset that is both closed and open, pick a point p that’s not in U and let x be the closest point to p in U, in the sense of length of path joining p (connectedness is used here). Then by closedness, x is a boundary point of U. However by openness of U there’s a neighborhood of x that lies inside U, which contradicts the choice of x since we can find another point that is closer to p in neighborhood of x. Note that closedness of U also guarantees that distance of p and x is positive, otherwise p will be a member of U.
This result is not true in general topological space that is not connected, say X = {1,2} with discrete topology.
Thank you for asking this , i couldn't find why for the life of me and thank you to the kind strangers for the answers1
Great Video, very informative.
Analytic continuation leads to the nightmare of Riemann surfaces.
Nightmare? A glimpse of a Garden of Eden, when compared with what a “raw” variety in R^n looks like on the average.
Riemann surfaces are some of the most beautiful mathematical objects :-)
@@antonios-alexandrosrobotis8205 These lectures prompted me to dig deeper, and I've just learned, that they can be approximated nicely by tropical algebra and tropical geometry. On the other hand one is rather troubled by young fast and furious fitting big data onto them.
Great video Thank you very much any video regarding the uniqueness of analytic continuation?
Are we going to get an updated version of locally uniform convergence?
Already here, sorry.
Entanglement?
Indeed that intro! It must be wrong! Convince me!
"really pedantic set theorist might object to this" lol
yeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
Please don't pollute a study space with comments like these. It's highly unprofessional. We should be thankful that Prof. Borcherds is spending so much of his time in the exposition of this material.
@@sjhd98 Apophis is just showing his excitement/appreciation for the content. This is a RUclips video, not a formal academic setting, and so some degree of informality is appropriate. I don't think there's any need to get all elitist about it.
@@conoroneill8067 FYI apophis comments something absolutely useless on _each_ and _every_ video Borcherds posts. It's bad.
@@ruinenlust_ It's harmless. Any perceived dearth of engaging comments is not due to useless comments 'polluting' the comment section of Borcherds' videos.
Yeeee (Translation: your "ye" comments became so familiar that they help to make it feel like we are classmates in Professor Borcherds' class)
yazın çok kötü reis
Sir, could you please go a little slower
You can just slow down the video.
The great thing about recorded lectures is you can pause, rewind, slow down, and ponder over particularly troubling sections for as long as you need.
terrible