Lightness of induced mappings

Charatonik, Janusz J.; Charatonik, Włodzimierz J.

doi:10.21099/tkbjm/1496163479

Cited by 21 publications

(12 citation statements)

References 3 publications

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“…The reader can find examples where the other implications are not true in [2]. Regarding the implications: (ii) implies (i) and (iii) implies (ii) the reader needs to use Theorem 3.5 to find the appropriate examples.…”

Section: Proposition 38 Let F : X → Y Be a Map Between Continua Conmentioning

confidence: 99%

“…In [2,Example 4.5], a map f between continua is given such that C( f ) is light, surjective and f is not monotone.…”

Section: Proposition 38 Let F : X → Y Be a Map Between Continua Conmentioning

confidence: 99%

“…Let f : X → Y be a map between continua. J. J. Charatonik and W. J. Charatonik [2] studied the relations between the following three statements: (i) f is light; (ii) C( f ) is light; (iii) 2 f is light.…”

Section: Introductionmentioning

confidence: 99%

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Lightness of Induced Maps and Homeomorphisms

Camargo¹

2011

Can. math. bull.

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Abstract. An example is given of a map f defined between arcwise connected continua such that C( f ) is light and 2 f is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive integer n, we study when the lightness of the induced map 2 f or Cn( f ) implies that f is a homeomorphism. Finally, we show a result in relation with the lightness of C(C( f )).

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Section: Proposition 38 Let F : X → Y Be a Map Between Continua Conmentioning

confidence: 99%

“…In [2,Example 4.5], a map f between continua is given such that C( f ) is light, surjective and f is not monotone.…”

Section: Proposition 38 Let F : X → Y Be a Map Between Continua Conmentioning

confidence: 99%

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Lightness of Induced Maps and Homeomorphisms

Camargo¹

2011

Can. math. bull.

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“…}, and note that f is well-defined. It has been proved in [1,Example 4.5] that the restriction C(f )|(C(X) \ {X}) is two-to-one and C(f ) −1 (X) is a singleton. Thus C(f ) is light and it is not a homeomorphism.…”

Section: Examplementioning

confidence: 99%

Openness and monotoneity of induced mappings

Charatonik¹

1999

Proc. Amer. Math. Soc.

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Abstract. It is shown that for locally connected continuum X if the induced mapping C(f ) : C(X) → C(Y ) is open, then f is monotone. As a corollary it follows that if the continuum X is hereditarily locally connected and C(f ) is open, then f is a homeomorphism. An example is given to show that local connectedness is essential in the result.All spaces considered in this paper are assumed to be metric. A mapping means a continuous function. We denote by N the set of all positive integers, and by C the complex plane. Given a space S, a point c ∈ S and a number ε > 0, we denote by B S (c, ε) the open ball in S with center c and radius ε.A continuum means a compact connected space. Given a continuum X with a metric d, we let 2 X denote the hyperspace of all nonempty closed subsets of X equipped with the Hausdorff metric H defined by H(A, B) = max{sup{d(a, B) : a ∈ A}, sup{d(b, A) : b ∈ B}}

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“…There are some papers in which particular results concerning this problem are shown for various classes M of mappings like open, monotone, confluent and some others, see [3,4,6,9,[11][12][13]17,22]. In the present paper we discuss the problem concerning possible relations between conditions (1.1)-(1.3) from one side, and the corresponding conditions in which an admissible class near-M, defined as the class of uniform limits of mappings belonging to M, from the other.…”

Section: Introductionmentioning

confidence: 99%