The Bolzano property and the cube-like complexes Przemys law Tkacz - PowerPoint PPT Presentation

The Bolzano property and the cube-like complexes Przemys� law Tkacz and Marian Turza´ nski Cardnial Stefan Wyszy´ nski University Warsaw, Poland

Introduction Theorem (Bolzano 1817) If a continuous f : [ a , b ] → R and f ( a ) · f ( b ) ≤ 0 , then there is c ∈ [ a , b ] such that f ( c ) = 0 .

Introduction Let I n = [0 , 1] n be an n -dimensional cube in R n . Its i-th opposite faces are defined as follows: i : = { x ∈ I n : x ( i ) = 0 } , I + i : = { x ∈ I n : x ( i ) = 1 } I −

Introduction Let I n = [0 , 1] n be an n -dimensional cube in R n . Its i-th opposite faces are defined as follows: i : = { x ∈ I n : x ( i ) = 0 } , I + i : = { x ∈ I n : x ( i ) = 1 } I −

Introduction Let I n = [0 , 1] n be an n -dimensional cube in R n . Its i-th opposite faces are defined as follows: i : = { x ∈ I n : x ( i ) = 0 } , I + i : = { x ∈ I n : x ( i ) = 1 } I −

Introduction Let I n = [0 , 1] n be an n -dimensional cube in R n . Its i-th opposite faces are defined as follows: i : = { x ∈ I n : x ( i ) = 0 } , I + i : = { x ∈ I n : x ( i ) = 1 } I −

The Poincar´ e-Miranda theorem Theorem (Poincar´ e 1883) If a continuous f = ( f 1 , f 2 , . . . , f n ) : I n → R n , f i ( I + f i ( I − i ) ⊂ ( −∞ , 0] , i ) ⊂ [0 , ∞ ) , then there is c ∈ I n such that f ( c ) = (0 , 0 , . . . , 0) .

The Poincar´ e-Miranda theorem Theorem (Poincar´ e 1883) If a continuous f = ( f 1 , f 2 , . . . , f n ) : I n → R n , f i ( I + f i ( I − i ) ⊂ ( −∞ , 0] , i ) ⊂ [0 , ∞ ) , then there is c ∈ I n such that f ( c ) = (0 , 0 , . . . , 0) . Theorem (Miranda 1940) The Poincar´ e theorem is equivalent to the Brouwer fixed point theorem.

The n -dimensional Bolzano property Definition (Kulpa 1994) The topological space X has the n-dimensional Bolzano property if there exists a family { ( A i , B i ) : i = 1 , . . . , n } of pairs of non-empty disjoint closed subsets such that for every continuous f = ( f 1 , . . . , f n ) : X → R n , for each i ≤ n f i ( A i ) ⊂ ( −∞ , 0] , and f i ( B i ) ⊂ [0 , ∞ ) , there exists c ∈ X such that f ( c ) = 0. { ( A i , B i ) : i = 1 , . . . , n } : an n-dimensional boundary system.

The n -dimensional Bolzano property Definition (Bolzano property) The topological space X has the n-dimensional Bolzano property if there exists a family { ( A i , B i ) : i = 1 , . . . , n } of pairs of disjoint closed subsets i , H + such that for every family { ( H − i ) : i = 1 , . . . , n } of closed sets such that for each i ≤ n i , B i ⊂ H + ∪ H + A i ⊂ H − and H − = X i i i we have � ∩ H + { H − : i = 1 , . . . , n } � = ∅ . i i

The n -dimensional Bolzano property Theorem If X has the n-dimensional Bolzano property. Then X has the Kulpa’s n-dimensional Bolzano property.

The n -dimensional Bolzano property Theorem If X has the n-dimensional Bolzano property. Then X has the Kulpa’s n-dimensional Bolzano property. Theorem If X is a normal and has the Kulpa’s n-dimensional Bolzano property. Than X has the n-dimensional Bolzano property.

Properties Theorem Let { ( A i , B i ) : i = 1 , ..., n } be the n-dimensional boundary system in T 5 space X. Then for each i 0 ∈ { 1 , . . . , n } A i 0 , B i 0 have an ( n − 1) -dimensional Bolzano property. Moreover the families { ( A i 0 ∩ A i , A i 0 ∩ B i ) : i � = i 0 } , { ( B i 0 ∩ A i , B i 0 ∩ B i ) : i � = i 0 } are an ( n − 1) -dimensional boundary systems in A i 0 , B i 0 respectively.

Properties Theorem Let { ( A i , B i ) : i = 1 , ..., n } be the n-dimensional boundary system in T 5 space X. Then for each i 0 ∈ { 1 , . . . , n } A i 0 , B i 0 have an ( n − 1) -dimensional Bolzano property. Moreover the families { ( A i 0 ∩ A i , A i 0 ∩ B i ) : i � = i 0 } , { ( B i 0 ∩ A i , B i 0 ∩ B i ) : i � = i 0 } are an ( n − 1) -dimensional boundary systems in A i 0 , B i 0 respectively. Corollary Let I 1 , I 2 ⊂ { 1 , . . . , n } , I 1 ∩ I 2 = ∅ . Then the subspace � � A i ∩ B i i ∈ I 1 i ∈ I 2 has an ( n − ( card ( I 1 ) + card ( I 2 ))) -dimensional Bolzano property.

An n -cube-like polyhedron Let A be a finite set. Definition All complexes consisting of a single vertex are 0-cube-like ( K 0 ). The complex K n generated by the family S ⊂ P n +1 ( A ) is said to be an n-cube-like complex if: (A) for every ( n − 1)-face T ∈ K n \ ∂ K n there exists exactly two n -simplexes S , S ′ ∈ K n such that S ∩ S ′ = T . i , F + (B) there exists a sequence of n pairs of subcomplexes F − called i i-th opposite faces such that: ( B 1 ) ∂ K n = � n ∪ F + i =1 F − i , i ∩ F + ( B 2 ) F − = ∅ for i ∈ { 1 , ..., n } , i i ( B 3 ) for each i 0 ∈ { 1 , ..., n } and each ǫ ∈ {− , + } , F ǫ i 0 is an ( n − 1)-cube-like complex such that its opposite faces have a form i 0 ∩ F + F ǫ i 0 ∩ F − i , F ǫ for i � = i 0 . i

An n -cube-like polyhedron

An n -cube-like polyhedron

An n -cube-like polyhedron

An n -cube-like polyhedron

An n -cube-like polyhedron

An n -cube-like polyhedron

An n -cube-like polyhedron Theorem Let ( ¯ K , ¯ K ) be an n-cube-like polyhedron in R m . Then ¯ K has an n-dimensional Bolzano property.

The Steinhaus chains Theorem (PT and Turza´ nski 2008) For an arbitrary decomposition of n-dimensional cube I n onto k n cubes and an arbitrary coloring function F : T ( k ) → { 1 , ... n } for some natural number i ∈ { 1 , ... n } there exists an i-th colored chain P 1 , ..., P r such that P 1 ∩ I + � = ∅ and P r ∩ I − � = ∅ . i i

The Steinhaus chains Theorem (PT and Turza´ nski 2008) For an arbitrary decomposition of n-dimensional cube I n onto k n cubes and an arbitrary coloring function F : T ( k ) → { 1 , ... n } for some natural number i ∈ { 1 , ... n } there exists an i-th colored chain P 1 , ..., P r such that P 1 ∩ I + � = ∅ and P r ∩ I − � = ∅ . i i ”1”-white ”2”-black

The Steinhaus chains Theorem (PT and Turza´ nski 2008) For an arbitrary decomposition of n-dimensional cube I n onto k n cubes and an arbitrary coloring function F : T ( k ) → { 1 , ... n } for some natural number i ∈ { 1 , ... n } there exists an i-th colored chain P 1 , ..., P r such that P 1 ∩ I + � = ∅ and P r ∩ I − � = ∅ . i i ”1”-white ”2”-black

The Steinhaus chains Theorem (PT and Turza´ nski 2008) For an arbitrary decomposition of n-dimensional cube I n onto k n cubes and an arbitrary coloring function F : T ( k ) → { 1 , ... n } for some natural number i ∈ { 1 , ... n } there exists an i-th colored chain P 1 , ..., P r such that P 1 ∩ I + � = ∅ and P r ∩ I − � = ∅ . i i ”1”-white ”2”-black

The Steinhaus chains Theorem (PT and Turza´ nski 2008) For an arbitrary decomposition of n-dimensional cube I n onto k n cubes and an arbitrary coloring function F : T ( k ) → { 1 , ... n } for some natural number i ∈ { 1 , ... n } there exists an i-th colored chain P 1 , ..., P r such that P 1 ∩ I + � = ∅ and P r ∩ I − � = ∅ . i i ”1”-white ”2”-black

The Steinhaus chains Theorem (PT and Turza´ nski 2008) For an arbitrary decomposition of n-dimensional cube I n onto k n cubes and an arbitrary coloring function F : T ( k ) → { 1 , ... n } for some natural number i ∈ { 1 , ... n } there exists an i-th colored chain P 1 , ..., P r such that P 1 ∩ I + � = ∅ and P r ∩ I − � = ∅ . i i Theorem (Topological version) Let { U i : i = 1 , · · · , n } be an open covering of I n . Then for some i ∈ { 1 , ... n } there exists continuum W ⊂ U i such that � = ∅ � = W ∩ I + W ∩ I − i . i

The Steinhaus chains Theorem (PT and Turza´ nski) The following statements are equivalent: 1. Theorem(on the existence of a chain) 2. The Poincar´ e theorem 3. The Brouwer Fixed Point theorem.

The Steinhaus chains Theorem (Michalik, P T, Turza´ nski 2015) Let K n be an n-cube-like complex. Then for every map φ : |K n | → { 1 , ..., n } there exist i ∈ { 1 , ..., n } and i-th colored chain { s 1 , ..., s m } ⊂ |K n | such that and s m ∈ F + s 1 ∈ F − i . i (The sequence { s 1 , ..., s m } ⊂ |K n | is a chain if for each i ∈ { 1 , ..., m − 1 } we have { s i , s i +1 } ∈ K n .)

Characterization of the Bolzano property Theorem Let X be a locally connected space. A family { ( A i , B i ) : i = 1 , . . . , n } of pairs of disjoint closed subsets is an n-dimensional boundary system iff for each open covering { U i } n i =1 for some i ≤ n there exists a connected set W ⊂ U i such that W ∩ A i � = ∅ � = W ∩ B i .

The inverse system Let us consider the inverse system { X σ , π σ ρ , Σ } where: (i) ∀ σ ∈ Σ X σ is a compact space with n -dimensional boundary system { ( A σ i , B σ i ) : i = 1 , ..., n } . (ii) ∀ σ, ρ ∈ Σ , ρ ≤ σ the map π σ ρ : X σ → X ρ is a surjection such that ρ ( A σ i ) = A ρ ρ ( B σ i ) = B ρ i . π σ i , π σ