Only 3-generalized metric spaces have a compatible symmetric topology

Tomonari Suzuki; Badriah Alamri; Misako Kikkawa

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 2301-2309
  • ISSN: 2391-5455

Abstract

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We prove that every 3-generalized metric space is metrizable. We also show that for any ʋ with ʋ ≥ 4, not every ʋ-generalized metric space has a compatible symmetric topology.

How to cite

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Tomonari Suzuki, Badriah Alamri, and Misako Kikkawa. "Only 3-generalized metric spaces have a compatible symmetric topology." Open Mathematics 13.1 (2015): 2301-2309. <http://eudml.org/doc/271766>.

@article{TomonariSuzuki2015,
abstract = {We prove that every 3-generalized metric space is metrizable. We also show that for any ʋ with ʋ ≥ 4, not every ʋ-generalized metric space has a compatible symmetric topology.},
author = {Tomonari Suzuki, Badriah Alamri, Misako Kikkawa},
journal = {Open Mathematics},
keywords = {ʋ-generalized metric space; Metrizability; Topology; Symmetrizable; Semimetrizable; fixed point; contraction; generalized metric space},
language = {eng},
number = {1},
pages = {2301-2309},
title = {Only 3-generalized metric spaces have a compatible symmetric topology},
url = {http://eudml.org/doc/271766},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Tomonari Suzuki
AU - Badriah Alamri
AU - Misako Kikkawa
TI - Only 3-generalized metric spaces have a compatible symmetric topology
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 2301
EP - 2309
AB - We prove that every 3-generalized metric space is metrizable. We also show that for any ʋ with ʋ ≥ 4, not every ʋ-generalized metric space has a compatible symmetric topology.
LA - eng
KW - ʋ-generalized metric space; Metrizability; Topology; Symmetrizable; Semimetrizable; fixed point; contraction; generalized metric space
UR - http://eudml.org/doc/271766
ER -

References

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  1. [1] B. Alamri, T. Suzuki and L. A. Khan, Caristi’s fixed point theorem and Subrahmanyam’s fixed point theorem in ʋ-generalized metric spaces, J. Funct. Spaces, 2015, Art. ID 709391, 6 pp. [WoS] Zbl1321.54055
  2. [2] A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31–37. MR1771669 Zbl0963.54031
  3. [3] G. Gruenhage, “Generalized metric spaces” in Handbook of set-theoretic topology, 1984, pp. 423–501, North-Holland, Amsterdam. MR0776629 
  4. [4] Z. Kadelburg and S. Radenovi´c, On generalized metric spaces: A survey, TWMS J. Pure Appl. Math., 5 (2014), 3–13. Zbl1305.54040
  5. [5] W. A. Kirk and N. Shahzad, Generalized metrics and Caristi’s theorem, Fixed Point Theory Appl., 2013, 2013:129. MR3068651 [WoS] 
  6. [6] T. Suzuki, Generalized metric spaces do not have the compatible topology, Abstr. Appl. Anal., 2014, Art. ID 458098, 5 pp. [WoS] 
  7. [7] S. Willard, General Topology, Dover (2004). MR2048350 

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