# Hereditarily weakly confluent induced mappings are homeomorphisms

Janusz Charatonik; Włodzimierz Charatonik

Colloquium Mathematicae (1998)

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

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

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