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### Connected Choice and the Brouwer Fixed Point Theorem [article]

Vasco Brattka, Stéphane Le Roux, Joseph S. Miller, Arno Pauly
2018 arXiv   pre-print
We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice.  ...  One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension.  ...  Brouwer's Fixed Point Theorem and Connected Choice In this section, we want to prove that the Brouwer Fixed Point Theorem is computably equivalent to connected choice for any fixed dimension.  ...

### Connected choice and the Brouwer fixed point theorem

Vasco Brattka, Stéphane Le Roux, Joseph S. Miller, Arno Pauly
2019 Journal of Mathematical Logic
Definition (Choice) The choice statement (∀ closed A ⊆ X )(∃x ∈ X )(A = ∅ =⇒ x ∈ A) translates into the choice operation C X :⊆ A − (X ) ⇒ X , A → A Realizing Theorems Definition Any theorem T of form  ...  By Fix n : C n → A n , f → {x ∈ [0, 1] n : f (x) = x} we denote the fixed point map of dimension n. We note that CC n • Con n • Fix n (f ) ⊆ BFT n (f ).  ...

### On the Computational Content of the Brouwer Fixed Point Theorem [chapter]

Vasco Brattka, Stéphane Le Roux, Arno Pauly
2012 Lecture Notes in Computer Science
Brouwer's Fixed Point Theorem and Connected Choice In this section we want to show that the Brouwer Fixed Point Theorem is computably equivalent to connected choice for any fixed dimension.  ...  It is easy to see that in general the Brouwer Fixed Point Theorem and connected choice are not independent of the dimension.  ...

### The Brouwer Fixed Point Theorem Revisited [chapter]

Vasco Brattka, Stéphane Le Roux, Joseph S. Miller, Arno Pauly
2016 Lecture Notes in Computer Science
In this paper we attempt to classify the Brouwer Fixed Point Theorem.  ...  One of our main results, presented in Section 3, is that the Brouwer Fixed Point Theorem is equivalent to connected choice for each fixed dimension n, i.e. BFT n ≡ W CC n .  ...

### The open mapping theorem and the fundamental theorem of algebra [article]

Daniel Reem
2011 arXiv   pre-print
In particular, this shows that one can obtain a proof of the FTA (albeit indirect) which is based on the Brouwer fixed point theorem, an aim which was not reached in the past and later the possibility  ...  In addition, it is also independent of the fact that the roots of an analytic function are isolated. The proof is based on either the Banach or Brouwer fixed point theorems.  ...  This cast doubt on the possibility of proving the FTA by applying the Brouwer fixed point theorem.  ...

### On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem

Yasuhito Tanaka
2006 Applied Mathematics and Computation
functions that there exists no social welfare function which satisfies transitivity, Pareto principle, independence of irrelevant alternatives, and has no dictator is equivalent to the Brouwer fixed point  ...  We will show that in the case where there are two individuals and three alternatives (or under the assumption of free-triple property) the Arrow impossibility theorem (Arrow (1963)) for social welfare  ...  to the Brouwer fixed point theorem.  ...

### On the Invariance of Openness

G. T. Whyburn
1958 Proceedings of the National Academy of Sciences of the United States of America
In the present theorem we keep the integers a and b fixed, but ki, k2, k3 are arbitrary integers with d any integer satisfying d 0 0 (mod p).  ...  and fixed integral exponents with x and y unknowns.  ...  In the present theorem we keep the integers a and b fixed, but ki, k2, k3 are arbitrary integers with d any integer satisfying d 0 0 (mod p).  ...

### Intersection properties of Helly families

2001 Topology and its Applications
The Helly convex-set theorem is extended onto topological spaces.  ...  From our results it follows that if there are given m + 2 convex subsets of an m-dimensional contractible Hausdorff space and the intersection of each collection of m + 1 the subsets is a nonempty contractible  ...  Our main tool is the Brouwer fixed point theorem (1912) which appears here in an equivalent form as lemma on indexed covering (some applications of the lemma to the fixed point theory are given in [6]  ...

### A FIXED-POINT THEOREM FOR PLANAR HOMEOMORPHISMS

Michael Handel
1999 Topology
Acknowledgement-I am grateful to the referee for a very careful reading of the manuscript and for many helpful suggestions that improved the exposition.  ...  INTRODUCTION The main result of this paper is the following fixed point theorem. THEOREM 2.3 .  ...  As before, fI "S is fixed point free and the Lefschetz Theorem implies that fI , and hence f, has a fixed point of positive index. This contradiction completes the proof of (a).  ...

### Monotone principle of forked points and its consequences

2015 Mathematica Moravica
Brouwer and Schauder theorems are two direct censequences of the monotone principle od forked points.  ...  Annalen, 65 (1908), 107-128; translated in van Heijenoort 1967, 183-198], and by M. Ta s k o v i ć [The axiom of choice, fixed point theorems, and inductive ordered sets, Proc. Amer. Math.  ...  We notice that this statement is a direct translation of the Brouwer fixed point theorem to Banach spaces. Proof.  ...

### THE HEX GAME THEOREM AND THE ARROW IMPOSSIBILITY THEOREM: THE CASE OF WEAK ORDERS

Yasuhito Tanaka
2009 Metroeconomica
Because Gale showed that the Brouwer fixed point theorem is equivalent to the HEX game theorem, this paper indirectly shows the equivalence of the Brouwer fixed point theorem and the Arrow impossibility  ...  Chichilnisky showed the equivalence of her impossibility theorem and the Brouwer fixed point theorem, and Baryshnikov showed that the impossibility theorem by Chichilnisky and the Arrow impossibility theorem  ...  CONCLUDING REMARKS We have considered the relationship between the HEX game theorem and the Arrow impossibility theorem when individual preferences are weak orders, and have shown their equivalence.  ...

### The Game of Hex and the Brouwer Fixed-Point Theorem

David Gale
1979 The American mathematical monthly
This research has been partially supported by the National Science Foundation under Grant MCS77-16054 and SOC78-05196 with the University of California, Berkeley.  ...  Reproduction in whole or in part is permitted for any purpose of the United States Government.  ...  We have thus obtained a fixed-point-free mapping, which contradicts the Brouwer Theorem. . 4 . 4 The n-dimensional Hex Theorem.  ...

### The Game of Hex and the Brouwer Fixed-Point Theorem

David Gale
1979 The American mathematical monthly
This research has been partially supported by the National Science Foundation under Grant MCS77-16054 and SOC78-05196 with the University of California, Berkeley.  ...  Reproduction in whole or in part is permitted for any purpose of the United States Government.  ...  We have thus obtained a fixed-point-free mapping, which contradicts the Brouwer Theorem. . 4 . 4 The n-dimensional Hex Theorem.  ...