Video: Heine Borel Theorem

Not yet rated
This video illustrates a proof of the Heine Borel Theorem: "Every closed bounded set in R^n is compact."





The proof goes like this:





Assume the contary. Let X be a bounded closed set (the disk in the video) that has an infinite open covering (symbolized by the colored small disks in the video) with no finite subcovering. Now devide X and the covering in two halfs. Now at least one sides does not admit a finite subcovering (otherwise we could construct a finite subcovering of X). Now subdivide the side without finite subcovering, and repeat the construction. We obain a sequence of subdivisions that do not have a finite subcovering of the given infinite covering.





If we now choose a point inside each of the subdivisions we obtain a cauchy sequence, which must have a limit point P since R^n is complete. P lies in every subdivision. Now choose an open set U (yellow in the movie) of the covering which contains P. Since U has a finite size it contains all subdivisions which are small enough. But then U is a finite subcovering for these subdivisions which contradicts our construction. Therefore our assumption that allowed the construction must be wrong and the Heine-Borel-Theorem correct. q.e.d








This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.





http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1





This film is #7 of our geometric animations calendar


http://www.calendar.algebraicsurface.net
Views: 2,820
Added: 10 years ago.

Share:

More Mathematics Videos

Visualization of Riemann Surfaces Visualization of Riemann Surfaces Heine Borel Theorem Heine Borel Theorem Vedic Mathematics: My Trip to India to Uncover the Truth - Alex Bellos (2011) Vedic Mathematics: My Trip to India to Uncover the Truth - Alex Bellos (2011) Early Mathematics: A Short Introduction - Professor Robin Wilson (2011) Early Mathematics: A Short Introduction - Professor Robin Wilson (2011) Mobius Story: Wind and Mr. Ug Mobius Story: Wind and Mr. Ug The Archimedes Codex - Professor Reviel Netz (2011) The Archimedes Codex - Professor Reviel Netz (2011)

Comments

There are no comments. Be the first to post one.
  Post comment as a guest user.
Click to login or register:
Your name:
Your email:
(will not appear)
Your comment:
(max. 1000 characters)
Are you human? (Sorry)