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Jun 28, 2023 35 likes •1.3k views

Colocally connected, Non-cut, Non- block and Shore sets in Hyperspaces and Symmetric Products Vernica Mar+nez de la Vega Jorge M Mar+nez Montejano XV Workshope on Con;nuum Theory and Dynamics Systems North Bay, ON 2018 Colocally connected,

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Colocally connected, Non-cut, Non- block and Shore sets in Hyperspaces and Symmetric Products

Verónica Mar+nez de la Vega Jorge M Mar+nez Montejano XV Workshope on Con;nuum Theory and Dynamics Systems North Bay, ON 2018

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Colocally connected, Non-cut, Non- block and shore sets in Hyperspaces and Symmetric Products

  • 1. Definitons
  • 2. Previous Results
  • 3 Main Result in the Hyperspace Cn(X)
  • 4. Main Results in Symmetric Products.
  • 5. Special Cases in Symmetric Products.
  • 6. Counter examples in Symmetric Products.
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  • 1. Defini;ons
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Defini;ons

  • A CONTINUUM is a compact

connected metric space.

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Defin;nions-Examples

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1.1. Hypersapaces

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HYPERSPACES

  • Given a con;nuum X, we define the

following hyperspaces:

  • 2X = { A ⊂ X : A≠Ǿ and A is closed }
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The Hyperspaces C(X) and Cn(X)

  • C(X) = { A Є 2X : A is connected }
  • Cn(X) = { A Є 2X : A has at most n

components }

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The Symmetric product Fn(X)

  • Fn(X) = { A Є 2X : A has at most n points }
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THE HAUSDORFF METRIC IN 2X

  • We endow 2X with the Hausdorff

metric H.

  • Since Fn(X), C(X) and Cn(X) are

subspaces of 2X, we endow them with the Hausdorff metric H, as well.

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1.3 Colocal connectedness

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Colocal connectedness

  • Let X be a con;nuum and A a

subcon;nuum of X with int(A) = ∅.

  • We say that A is a con;nuum of colocal

connectedness in X provided that for each open subset U of X with A ⊂ U there exists an open subset V of X such that A ⊂ V ⊂ U and X \ V is connected.

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1.4 Not a weak cut con;nuum

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Not a weak cut

  • Let X be a con;nuum and A a

subcon;nuum of X with int(A) = ∅. We say that A is not a weak cut con;nuum in X if for any pair of points x,y ∈ X / A there is a subcon;nuum M of X such that x, y ∈ M and M ∩ A = ∅.

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1.5 Non Block

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Non Block

  • Let X be a con;nuum and A a subcon;nuum
  • f X with int(A) = ∅.

We say that A is a nonblock con;nuum in X provided that there exist a sequence of subcon;nua M1,M2,... such that M1 ⊂ M2 ⊂ · · · and UMn is dense subset of X \ A.

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1.6 Shore, Not strong Center and Non Cut

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Shore, Not strong Center and Non Cut

  • Let X be a con;nuum and A a

subcon;nuum of X with int(A) = ∅. We say that A is: a shore con=nuum in X if for each ε > 0 there is a subcon;nuum M of X such that H(M, X) < ε and M ∩ A = ∅.

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Shore, Not strong Center and Non Cut

  • (5) not a strong center in X provided

that for each pair of nonempty open subsets U and V of X there exists a subcon;nuum M of X such that M∩U ≠ ∅ ≠ M∩V and M∩A = ∅.

  • (6) a noncut con;nuum in X if X \ A

is connected.

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  • 2. Previous Results
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Theorem

  • J. Bobok, P. Pyrih and B. Vejnar
  • Colocally Connected => non- weak cut =>

non-block => shore => strong center => non cut

  • Non cut ≠>strong center ≠>shore≠>non block

≠> non weak cut ≠> colocally connected

  • If X is locally connected then Colocally

Connected <=> non- weak cut <=> non-block <=> shore< => strong center< => non cut

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F1(X) is a sumbcon;nuum in H(X)

  • Where H(X) is any Hyperspace,

H(X) Є{ 2X, Fn(X), C(X), Cn(X) } In fact F1(X) has empty interior in H(X) F1(X) is homeomorphic to X.

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So we want to know if F1(X) is a :

  • Collocal connected
  • Non weak cut
  • Non block
  • Shore
  • Strong center
  • Non cut

subcon;num in H(X).

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3 Main Result in the Hyperspace Cn(X)

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Theorem (VMV and JMM, 2017)

  • For every con;nuum X and each

posi;ve integer n, F1(X) is a colocally connected subcon;nuum in Cn(X)

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ORDERED ARCS IN HYPERSPACES

  • Given the Hyperspace Cn(X) if A,B Є

Cn(X) and A ⊂ B we define an

  • rdered arc from A to B in Cn(X) is a

map α:[0,1]àCn(X) such that:

  • α(0)=A, α(1)=B and
  • if 0 ≤ s < t ≤ 1 then α(s) ⊂α(t)
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Theorem (VMV and JMM, 2017)

  • For every con;nuum X and each

posi;ve integer n, F1(X) is a colocally connected subcon;nuum in 2X

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4 Main Results in Symmetric Products.

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Theorem (VMV and JMM, 2017)

  • For every con;nuum X and each

posi;ve integer n ≥ 3, F1(X) is a colocally connected set in Fn(X)

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Theorem (VMV and JMM, 2017)

  • For every locally connected

con;nuum X and each integer n, n = 2, F1(X) is a colocally connected set in Fn(X)

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Theorem (VMV and JMM, 2017)

  • Let X be a con;nuum and n = 2.

Then F1(X) is a nonblock con;nuum in Fn(X).

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Theorem (VMV and JMM, 2017)

  • The Knaster con;nuum K sa;sfies

that F1(K) is not colocally connected in F2(X)

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Theorem (VMV and JMM, 2017)

  • Let X be the sin 1/x con;nuum,

then F1(X) is a weak cut con;nuum in F2(X).

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  • Theorem (2018 VMV & JMMM)

Let X be a nonlocally connected chainable con;nuum. Suppose that there is a monotone map π : X à[0,1] such that for each a,b Є π -1([a,b]) with a < b. Then F1(X) is a weak cut con;nuum in F2(X).

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Theorem (VMV and JMM, 2017)

  • For each arcwise connected

con;nuum X and for n = 2, F1(X) is a non-cut set in Fn(X)

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Theorem (VMV and JMM, 2017)

  • There exists a dendroid X such that

F1(X) is not a colocally connected set in F2(X).

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Colocally connected, Non-cut, Non- block and shore sets in Hyperspaces and Symmetric Products

  • 1. For every con;nuum X and each

integer n, n ≥ 2, F1(X) is a colocal connected set in Cn(X)

  • 2. For every locally connected

con;nuum X and each posi;ve integer n, F1(X) is a colocal connected set in Fn(X)

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Colocally connected, Non-cut, Non- block and shore sets in Hyperspaces and Symmetric Products

  • 3. For each con;nuum X and posi;ve

integer n, F1(X) is a non-block con;nuum in Fn(X)

  • 4. The Knaster con;nuum K sa;sfies

that F1(K) is not colocally connected in F2(X)

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Colocally connected, Non-cut, Non- block and shore sets in Hyperspaces and Symmetric Products

  • 5. The sin(1/x) curve S sa;sfies that

F1(s) is a weak cut con;nuum in F2(X)

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Colocally connected, Non-cut, Non-block and shore sets in Hyperspaces and Symmetric Products

  • 8. For each arcwise connected

con;nuum X, F1(X) is a non-cut set in Fn(X)

  • 9. There exists a dendroid X such that

F1(X) is not a colocally connected set in F2(X).

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