Setwise and Pointwise Betweenness via Hyperspaces Qays Shakir joint - PowerPoint PPT Presentation

Setwise and Pointwise Betweenness via Hyperspaces Qays Shakir joint with Aisling McCluskey National University of Ireland,Galway 12th Symposium on General Topology and its Relations to Modern Analysis and Algebra 25 − 29 July 2016 Prague, Czech Republic

Overview Motivation 1 Setwise Betweenness via 2 X and F n ( X ) 2 Pointwise Betweenness via 2 X and F n ( X ) 3

Motivation An intuitive view of betweenness arises naturally in any order-theoretic structure; given a preorder ≤ on a set X , with a , b , c ∈ X such that a ≤ c ≤ b , we say that ” c is between a and b ”. Let ( X , ≤ ) be a partially ordered set. Define for a ≤ b , [ a , b ] O = { c ∈ X : a ≤ c ≤ b } . If ( X , ≤ ) is a tree with a common lower bound d of a , b . O ( a , b , d ) = [ d , a ] O � [ d , b ] O . b a

Motivation An intuitive view of betweenness arises naturally in any order-theoretic structure; given a preorder ≤ on a set X , with a , b , c ∈ X such that a ≤ c ≤ b , we say that ” c is between a and b ”. Let ( X , ≤ ) be a partially ordered set. Define for a ≤ b , [ a , b ] O = { c ∈ X : a ≤ c ≤ b } . If ( X , ≤ ) is a tree with a common lower bound d of a , b . Define O ( a , b , d ) = [ d , a ] O � [ d , b ] O . Define [ a , b ] T = { c ∈ O ( a , b , d ) : d ≤ a , b } d d d b a

Motivation An intuitive view of betweenness arises naturally in any order-theoretic structure; given a preorder ≤ on a set X , with a , b , c ∈ X such that a ≤ c ≤ b , we say that ” c is between a and b ”. Let ( X , ≤ ) be a partially ordered set. Define for a ≤ b , [ a , b ] O = { c ∈ X : a ≤ c ≤ b } . If ( X , ≤ ) is a tree with a common lower bound d of a , b . Define O ( a , b , d ) = [ d , a ] O � [ d , b ] O . Define [ a , b ] T = { c ∈ O ( a , b , d ) : d ≤ a , b } d d d c c c b a

Motivation Let X be a vector space over the real field R and let a , b ∈ X . The convex interval can be defined as follows: A vector c ∈ X is between a and b if c ∈ [ a , b ] conv = { at + (1 − t ) b : t ∈ [0 , 1] } . So [ a , b ] conv is the set of all convex combinations of a and b . b a o

Motivation Let X be a vector space over the real field R and let a , b ∈ X . The convex interval can be defined as follows: A vector c ∈ X is between a and b if c ∈ [ a , b ] conv = { at + (1 − t ) b : t ∈ [0 , 1] } . So [ a , b ] conv is the set of all convex combinations of a and b . b a c O

Road Systems and Pointwise Betweenness Paul Bankston introduced the following definitions: Definition A road system is a pair � X , R� , where X is a nonempty set and R is a collection of nonempty subsets of X -called the roads- such that: 1 For each a ∈ X , the singleton set { a } is a road. 2 Each two points a , b ∈ X belong to at least one road.

Road Systems and Pointwise Betweenness Paul Bankston introduced the following definitions: Definition A road system is a pair � X , R� , where X is a nonempty set and R is a collection of nonempty subsets of X -called the roads- such that: 1 For each a ∈ X , the singleton set { a } is a road. 2 Each two points a , b ∈ X belong to at least one road. Definition Let � X , R� be a road system and a , b , c ∈ X . Then c ∈ [ a , b ] R if every road containing a and b also contain c . Then c ∈ [ a , b ] R if c ∈ � { R ∈ R : R ∈ R ( a , b ) } where R ( a , b ) denotes the set of roads that contain both a and b

Road Systems and Setwise Betweenness There is a natural generalisation from pointwise betweenness to setwise betweenness as follows: Definition Let � X , R� be a road system with a , b ∈ X and ∅ � = C ⊆ X . We say that C is between a and b if C � R � = ∅ for all R ∈ R ( a , b )

Vietoris Hyperspace (2 X ) Definition Let X be a T 1 space. The Vietoris topology 2 X on CL ( X ), the collection of all non-empty closed subsets of X , is the one generated by sets of the form U + = { A ∈ CL ( X ) : A ⊂ U } U − = { A ∈ CL ( X ) : A � U � = ∅} where U is an open subset of X . A basis of the Vietoris topology consists of the collection of sets of the form n � � � U 1 , U 2 , ..., U n � = { A ∈ CL ( X ) : A ⊆ U i and if i ≤ n , A U i � = ∅} i =1 where U 1 , U 2 , ..., U n are non-empty open subsets of X .

n -fold Symmetric Product hyperspace F n ( X ) Definition Let X be a T 1 space, the hyperspace F n ( X ), called n -fold symmetric product of X , is a subspace of the Vietoris space 2 X defined as follows F n ( X ) = { A ∈ X : | A | ≤ n } Some properties of F n ( X ) F 1 ( X ) ∼ = X F n ( X ) ⊆ F n +1 ( X ) If X is a Hausdorff space then F n ( X ) is a closed subspace in the Vietoris hyperspace.

Setwise Betweenness via 2 X and F n ( X )

Setwise Betweenness via 2 X and F n ( X ) Notation Let X be a topological space and a , b ∈ X , the collection of sets that satisfies a topological property P forms a road system. The collection of sets that contain a and b and satisfy a topological property P is denoted by P ( a , b ). Definition Let X be a T 1 space. Define the setwise interval with respect to a property P and a hyperspace H as follows: H = { C ∈ H : C � K � = ∅ for every K ∈ P ( a , b ) } [ a , b ] S P

The Setwise Interval [ a , b ] S CO 2 X

The Setwise Interval [ a , b ] S CO 2 X Definition Let X be a topological space. Define the setwise interval with respect to the Vietoris hyperspace 2 X as follows: = { C ∈ 2 X : C � K � = ∅ for every K ∈ CO ( a , b ) } [ a , b ] S CO 2 X where CO ( a , b ) the collection of all connected sets that contains a and b .

The Setwise Interval [ a , b ] S CO 2 X Example Let X = C � B be a subspace of the R 2 where C = [ 1 2 , 1] and B = { (0 , 0) } � ∞ n =1 C n . Now if a ∈ C i and b ∈ C j with i � = j then for a A ∈ 2 X to be lie in the interval [ a , b ] S CO it is necessary and sufficient that 2 X (0 , 0) ∈ A .

Some Properties of The Interval [ a , b ] S CO 2 X Let X be a T 1 space with a , b ∈ X . Then 1 { a } , { b } ∈ [ a , b ] S CO 2 X 2 [ a , b ] S CO ⊆ [ a , a ] S CO , [ b , b ] S CO 2 X 2 X 2 X

Some Properties of The Interval [ a , b ] S CO 2 X Let X be a topological space with a , b ∈ X . Then 1 { a } , { b } ∈ [ a , b ] S CO 2 X 2 [ a , b ] S CO ⊆ [ a , a ] S CO , [ b , b ] S CO 2 X 2 X 2 X Theorem If f : X − → Y be a homeomorphism then f ([ a , b ] S CO ) = [ f ( a ) , f ( b )] S CO 2 X 2 Y

The Setwise Interval [ a , b ] S CO n ( X )

The Setwise Interval [ a , b ] S CO n ( X ) Definition Let X be a topological space. Define the setwise interval with respect to the n-fold symmetric product hyperspace F n ( X ) as follows: n ( X ) = { C ∈ F n ( X ) : C � K � = ∅ for every K ∈ CO ( a , b ) } [ a , b ] S CO

The Setwise Interval [ a , b ] S CO n ( X ) Example Let X be the comb space and A = { [ x , 0] � [0 . 2 , y ] : where 0 . 2 ≤ x ≤ 0 . 6 and 0 ≤ y ≤ 0 . 4 } . It is clear that A ∈ CO ( a , b ). Thus for C ∈ F n ( X ) to lie between a and b , i.e. to be sure that C ∈ [ a , b ] P CO n ( X ) it is enough for C to intersect A .

Some Properties of The Interval [ a , b ] S CO n ( X ) continue .... Some properties of the setwise interval [ a , b ] S CO n ( X ) Let X be a topological space with a , b ∈ X . Then 1 { a } , { b } ∈ [ a , b ] S CO n ( X ) 2 [ a , b ] S CO n ( X ) ⊆ [ a , a ] S CO n ( X ) , [ b , b ] S CO n ( X ) 3 For n ≥ 3, we have [ a , b ] S CO � [ b , c ] S CO n ( X ) � = ∅ n ( X ) 4 [ a , b ] S CO 1( X ) ⊆ [ a , b ] S CO 2( X ) ⊆ ... ⊆ [ a , b ] S CO n ( X )

Some Properties of The Interval [ a , b ] S CO n ( X ) continue .... Some properties of the setwise interval [ a , b ] S CO n ( X ) Let X be a topological space with a , b ∈ X . Then 1 { a } , { b } ∈ [ a , b ] S CO n ( X ) 2 [ a , b ] S CO n ( X ) ⊆ [ a , a ] S CO n ( X ) , [ b , b ] S CO n ( X ) 3 For n ≥ 3, we have [ a , b ] S CO � [ b , c ] S CO n ( X ) � = ∅ n ( X ) 4 [ a , b ] S CO 1( X ) ⊆ [ a , b ] S CO 2( X ) ⊆ ... ⊆ [ a , b ] S CO n ( X ) Proposition Proposition: Let X be a topological space with a , b ∈ X and C i ∈ F n ( X ) for i = 1 , 2 , ... such that C 1 ⊂ C 2 ⊂ ... . If C 1 ∈ [ a , b ] S CO n ( X ) then C i ∈ [ a , b ] S CO n ( X ) for each i = 2 , 3 , ...

Some Properties of The Interval [ a , b ] S CO n ( X ) continue .... Theorem If f : X − → Y be a homeomorphism then f ([ a , b ] S CO n ( X ) ) = [ f ( a ) , f ( b )] S CO n ( Y )

Pointwise Betweenness via 2 X and F n ( X ) Definition Let X be a topological space with x ∈ X . The hyperstar collection of x with respect to a hyperspace H is defined by st ( x , H ) = { C ∈ H : x ∈ C }

Pointwise Betweenness via 2 X and F n ( X ) Definition Let X be a topological space with x ∈ X . We define the hyperstar collection of x with respect to a hyperspace H as follows: st ( x , H ) = { C ∈ H : x ∈ C } st ( x , 2 X ) = { C ∈ 2 X : x ∈ C } st ( x , F n ( X )) = { C ∈ F n ( X ) : x ∈ C }

Pointwise Betweenness via 2 X and F n ( X ) Definition Let X be a topological space with x ∈ X . We define the hyperstar collection of x with respect to a hyperspace H is defined by st ( x , H ) = { C ∈ H : x ∈ C } st ( x , 2 X ) = { C ∈ 2 X : x ∈ C } st ( x , F n ( X )) = { C ∈ F n ( X ) : x ∈ C } Some properties of st ( x , F n ( X )) st ( x , F 1 ( X )) = {{ x }} st ( x , F n ( X )) ⊂ st ( x , F n +1 ( X ))