Brouwer fixed-point theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 28/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem


00:03:06 1 Statement
00:04:21 2 Importance of the pre-conditions
00:04:48 2.1 Boundedness
00:05:40 2.2 Closedness
00:06:35 2.3 Convexity
00:06:38 2.4 Notes
00:08:38 3 Illustrations
00:08:55 4 Intuitive approach
00:11:16 4.1 Explanations attributed to Brouwer
00:11:26 4.2 One-dimensional case
00:13:01 5 History
00:14:30 5.1 Prehistory
00:14:53 5.2 First proofs
00:15:38 5.3 Reception
00:19:11 6 Proof outlines
00:22:23 6.1 A proof using degree
00:26:21 6.2 A proof using homology
00:26:31 6.3 A proof using Stokes's theorem
00:34:29 6.4 A combinatorial proof
00:35:52 6.5 A proof by Hirsch
00:37:14 6.6 A proof using the ioriented area/i
00:38:30 6.7 A proof using the game hex
00:40:41 6.8 A proof using the Lefschetz fixed-point theorem
00:45:13 6.9 A proof in a weak logical system
00:46:55 7 Generalizations
00:49:31 8 Equivalent results
00:50:07 9 See also
00:51:34 10 Notes
00:52:04 11 References
00:56:30 12 External links
00:57:11 See also



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SUMMARY
=======
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function 
  
    
      
        f
      
    
    {\displaystyle f}
   mapping a compact convex set to itself there is a point 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x_{0}}
   such that 
  
    
      
        f
        (
        
          x
          
            0
          
        
        )
        =
        
          x
          
            0
          
        
      
    
    {\displaystyle f(x_{0})=x_{0}}
  . The simplest forms of Brouwer's theorem are for continuous functions 
  
    
      
        f
      
    
    {\displaystyle f}
   from a closed interval 
  
    
      
        I
      
    
    {\displaystyle I}
   in the real numbers to itself or from a closed disk 
  
    
      
        D
      
    
    {\displaystyle D}
   to itself. A more general form than the latter is for continuous functions from a convex compact subset 
  
    
      
        K
      
    
    {\displaystyle K}
   of Euclidean space to itself.
Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics.
In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.
This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.
It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.
The theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Jacques Hadamard and by Luitzen Egbertus Jan Brouwer.