Hereditarily weakly confluent induced mappings are homeomorphisms

Janusz Charatonik; Włodzimierz Charatonik

Colloquium Mathematicae (1998)

  • Volume: 75, Issue: 2, page 195-203
  • ISSN: 0010-1354

Abstract

top
For a given mapping f between continua we consider the induced mappings between the corresponding hyperspaces of closed subsets or of subcontinua. It is shown that if either of the two induced mappings is hereditarily weakly confluent (or hereditarily confluent, or hereditarily monotone, or atomic), then f is a homeomorphism, and consequently so are both the induced mappings. Similar results are obtained for mappings between cones over the domain and over the range continua.

How to cite

top

Charatonik, Janusz, and Charatonik, Włodzimierz. "Hereditarily weakly confluent induced mappings are homeomorphisms." Colloquium Mathematicae 75.2 (1998): 195-203. <http://eudml.org/doc/210538>.

@article{Charatonik1998,
abstract = {For a given mapping f between continua we consider the induced mappings between the corresponding hyperspaces of closed subsets or of subcontinua. It is shown that if either of the two induced mappings is hereditarily weakly confluent (or hereditarily confluent, or hereditarily monotone, or atomic), then f is a homeomorphism, and consequently so are both the induced mappings. Similar results are obtained for mappings between cones over the domain and over the range continua.},
author = {Charatonik, Janusz, Charatonik, Włodzimierz},
journal = {Colloquium Mathematicae},
keywords = {atriodic; semi-confluent; hereditary; confluent; atomic; homeomorphism; cone; weakly confluent; monotone; hyperspace; joining; continuum; confluent mapping},
language = {eng},
number = {2},
pages = {195-203},
title = {Hereditarily weakly confluent induced mappings are homeomorphisms},
url = {http://eudml.org/doc/210538},
volume = {75},
year = {1998},
}

TY - JOUR
AU - Charatonik, Janusz
AU - Charatonik, Włodzimierz
TI - Hereditarily weakly confluent induced mappings are homeomorphisms
JO - Colloquium Mathematicae
PY - 1998
VL - 75
IS - 2
SP - 195
EP - 203
AB - For a given mapping f between continua we consider the induced mappings between the corresponding hyperspaces of closed subsets or of subcontinua. It is shown that if either of the two induced mappings is hereditarily weakly confluent (or hereditarily confluent, or hereditarily monotone, or atomic), then f is a homeomorphism, and consequently so are both the induced mappings. Similar results are obtained for mappings between cones over the domain and over the range continua.
LA - eng
KW - atriodic; semi-confluent; hereditary; confluent; atomic; homeomorphism; cone; weakly confluent; monotone; hyperspace; joining; continuum; confluent mapping
UR - http://eudml.org/doc/210538
ER -

References

top
  1. [1] R. D. Anderson, Atomic decompositions of continua, Duke Math. J. 24 (1956), 507-514. Zbl0073.39701
  2. [2] J. J. Charatonik and W. J. Charatonik, Lightness of induced mappings, Tsukuba J. Math., to appear. Zbl0939.54005
  3. [3] W. J. Charatonik, Arc approximation property and confluence of induced mappings, Rocky Mountain J. Math., to appear. Zbl0926.54024
  4. [4] A. Emeryk and Z. Horbanowicz, On atomic mappings, Colloq. Math. 27 (1973), 49-55. 
  5. [5] H. Hosokawa, Some remarks on the atomic mappings, Bull. Tokyo Gakugei Univ. (4) 40 (1988), 31-37. 
  6. [6] H. Hosokawa, Induced mappings between hyperspaces, ibid. 41 (1989), 1-6. 
  7. [7] H. Hosokawa, Induced mappings between hyperspaces II, ibid. 44 (1992), 1-7. Zbl0767.54005
  8. [8] H. Hosokawa, Induced mappings on hyperspaces, Tsukuba J. Math., to appear. 
  9. [9] H. Hosokawa, Induced mappings on hyperspaces II, ibid., to appear. 
  10. [10] A. Y. W. Lau, A note on monotone maps and hyperspaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 121-123. Zbl0319.54011
  11. [11] T. Maćkowiak, Continuous mappings on continua, Dissertationes Math. (Rozprawy Mat.) 158 (1979). Zbl0444.54021
  12. [12] S. B. Nadler, Jr., Hyperspaces of Sets, Dekker, 1978. 

NotesEmbed ?

top

You must be logged in to post comments.