Balls in generalizations of metric spaces

Abstract This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let ( X , p b ) $(X,p_{b})$ be a partial b-metric space in the sense of Mustafa et al. For the family △ of all p b $p_{b}$ -open balls in ( X , p b ) $(X,p_{b})$ , this paper proves that there are x...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Xun Ge [verfasserIn] Shou Lin [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2016

Schlagwörter:	ball partial b-metric space cone metric space

Übergeordnetes Werk:	In: Journal of Inequalities and Applications - SpringerOpen, 2002, (2016), 1, Seite 7
Übergeordnetes Werk:	year:2016 ; number:1 ; pages:7

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.1186/s13660-016-0962-y

Katalog-ID:	DOAJ043357814

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520			\|a Abstract This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let ( X , p b ) $(X,p_{b})$ be a partial b-metric space in the sense of Mustafa et al. For the family △ of all p b $p_{b}$ -open balls in ( X , p b ) $(X,p_{b})$ , this paper proves that there are x , y ∈ B ∈ △ $x,y\in B\in\triangle$ such that B ′ ⊈ B $B'\nsubseteq B$ for all B ′ ∈ △ $B'\in\triangle$ , where B and B ′ $B'$ are with centers x and y, respectively. This result shows that △ is not a base of any topology on X, which shows that a proposition and a claim on partial b-metric spaces are not true. By some relations among ≪, <, and ≤ in cone metric spaces, this paper also constructs a cone metric space ( X , d ) $(X,d)$ and shows that { y ∈ X : d ( x , y ) ≪ ε } ‾ ≠ { y ∈ X : d ( x , y ) ≤ ε } $\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le\varepsilon\}$ in general, which corrects an error on cone metric spaces. However, it must be emphasized that these corrections do not affect the rest of the results in the relevant papers.
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spelling	10.1186/s13660-016-0962-y doi (DE-627)DOAJ043357814 (DE-599)DOAJefc111cd81694827ab68106af9e532cf DE-627 ger DE-627 rakwb eng QA1-939 Xun Ge verfasserin aut Balls in generalizations of metric spaces 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let ( X , p b ) $(X,p_{b})$ be a partial b-metric space in the sense of Mustafa et al. For the family △ of all p b $p_{b}$ -open balls in ( X , p b ) $(X,p_{b})$ , this paper proves that there are x , y ∈ B ∈ △ $x,y\in B\in\triangle$ such that B ′ ⊈ B $B'\nsubseteq B$ for all B ′ ∈ △ $B'\in\triangle$ , where B and B ′ $B'$ are with centers x and y, respectively. This result shows that △ is not a base of any topology on X, which shows that a proposition and a claim on partial b-metric spaces are not true. By some relations among ≪, <, and ≤ in cone metric spaces, this paper also constructs a cone metric space ( X , d ) $(X,d)$ and shows that { y ∈ X : d ( x , y ) ≪ ε } ‾ ≠ { y ∈ X : d ( x , y ) ≤ ε } $\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le\varepsilon\}$ in general, which corrects an error on cone metric spaces. However, it must be emphasized that these corrections do not affect the rest of the results in the relevant papers. ball partial b-metric space cone metric space Mathematics Shou Lin verfasserin aut In Journal of Inequalities and Applications SpringerOpen, 2002 (2016), 1, Seite 7 (DE-627)320977056 (DE-600)2028512-7 1029242X nnns year:2016 number:1 pages:7 https://doi.org/10.1186/s13660-016-0962-y kostenfrei https://doaj.org/article/efc111cd81694827ab68106af9e532cf kostenfrei http://link.springer.com/article/10.1186/s13660-016-0962-y kostenfrei https://doaj.org/toc/1029-242X Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2016 1 7
allfields_unstemmed	10.1186/s13660-016-0962-y doi (DE-627)DOAJ043357814 (DE-599)DOAJefc111cd81694827ab68106af9e532cf DE-627 ger DE-627 rakwb eng QA1-939 Xun Ge verfasserin aut Balls in generalizations of metric spaces 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let ( X , p b ) $(X,p_{b})$ be a partial b-metric space in the sense of Mustafa et al. For the family △ of all p b $p_{b}$ -open balls in ( X , p b ) $(X,p_{b})$ , this paper proves that there are x , y ∈ B ∈ △ $x,y\in B\in\triangle$ such that B ′ ⊈ B $B'\nsubseteq B$ for all B ′ ∈ △ $B'\in\triangle$ , where B and B ′ $B'$ are with centers x and y, respectively. This result shows that △ is not a base of any topology on X, which shows that a proposition and a claim on partial b-metric spaces are not true. By some relations among ≪, <, and ≤ in cone metric spaces, this paper also constructs a cone metric space ( X , d ) $(X,d)$ and shows that { y ∈ X : d ( x , y ) ≪ ε } ‾ ≠ { y ∈ X : d ( x , y ) ≤ ε } $\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le\varepsilon\}$ in general, which corrects an error on cone metric spaces. However, it must be emphasized that these corrections do not affect the rest of the results in the relevant papers. ball partial b-metric space cone metric space Mathematics Shou Lin verfasserin aut In Journal of Inequalities and Applications SpringerOpen, 2002 (2016), 1, Seite 7 (DE-627)320977056 (DE-600)2028512-7 1029242X nnns year:2016 number:1 pages:7 https://doi.org/10.1186/s13660-016-0962-y kostenfrei https://doaj.org/article/efc111cd81694827ab68106af9e532cf kostenfrei http://link.springer.com/article/10.1186/s13660-016-0962-y kostenfrei https://doaj.org/toc/1029-242X Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2016 1 7
allfieldsGer	10.1186/s13660-016-0962-y doi (DE-627)DOAJ043357814 (DE-599)DOAJefc111cd81694827ab68106af9e532cf DE-627 ger DE-627 rakwb eng QA1-939 Xun Ge verfasserin aut Balls in generalizations of metric spaces 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let ( X , p b ) $(X,p_{b})$ be a partial b-metric space in the sense of Mustafa et al. For the family △ of all p b $p_{b}$ -open balls in ( X , p b ) $(X,p_{b})$ , this paper proves that there are x , y ∈ B ∈ △ $x,y\in B\in\triangle$ such that B ′ ⊈ B $B'\nsubseteq B$ for all B ′ ∈ △ $B'\in\triangle$ , where B and B ′ $B'$ are with centers x and y, respectively. This result shows that △ is not a base of any topology on X, which shows that a proposition and a claim on partial b-metric spaces are not true. By some relations among ≪, <, and ≤ in cone metric spaces, this paper also constructs a cone metric space ( X , d ) $(X,d)$ and shows that { y ∈ X : d ( x , y ) ≪ ε } ‾ ≠ { y ∈ X : d ( x , y ) ≤ ε } $\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le\varepsilon\}$ in general, which corrects an error on cone metric spaces. However, it must be emphasized that these corrections do not affect the rest of the results in the relevant papers. ball partial b-metric space cone metric space Mathematics Shou Lin verfasserin aut In Journal of Inequalities and Applications SpringerOpen, 2002 (2016), 1, Seite 7 (DE-627)320977056 (DE-600)2028512-7 1029242X nnns year:2016 number:1 pages:7 https://doi.org/10.1186/s13660-016-0962-y kostenfrei https://doaj.org/article/efc111cd81694827ab68106af9e532cf kostenfrei http://link.springer.com/article/10.1186/s13660-016-0962-y kostenfrei https://doaj.org/toc/1029-242X Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2016 1 7
allfieldsSound	10.1186/s13660-016-0962-y doi (DE-627)DOAJ043357814 (DE-599)DOAJefc111cd81694827ab68106af9e532cf DE-627 ger DE-627 rakwb eng QA1-939 Xun Ge verfasserin aut Balls in generalizations of metric spaces 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let ( X , p b ) $(X,p_{b})$ be a partial b-metric space in the sense of Mustafa et al. For the family △ of all p b $p_{b}$ -open balls in ( X , p b ) $(X,p_{b})$ , this paper proves that there are x , y ∈ B ∈ △ $x,y\in B\in\triangle$ such that B ′ ⊈ B $B'\nsubseteq B$ for all B ′ ∈ △ $B'\in\triangle$ , where B and B ′ $B'$ are with centers x and y, respectively. This result shows that △ is not a base of any topology on X, which shows that a proposition and a claim on partial b-metric spaces are not true. By some relations among ≪, <, and ≤ in cone metric spaces, this paper also constructs a cone metric space ( X , d ) $(X,d)$ and shows that { y ∈ X : d ( x , y ) ≪ ε } ‾ ≠ { y ∈ X : d ( x , y ) ≤ ε } $\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le\varepsilon\}$ in general, which corrects an error on cone metric spaces. However, it must be emphasized that these corrections do not affect the rest of the results in the relevant papers. ball partial b-metric space cone metric space Mathematics Shou Lin verfasserin aut In Journal of Inequalities and Applications SpringerOpen, 2002 (2016), 1, Seite 7 (DE-627)320977056 (DE-600)2028512-7 1029242X nnns year:2016 number:1 pages:7 https://doi.org/10.1186/s13660-016-0962-y kostenfrei https://doaj.org/article/efc111cd81694827ab68106af9e532cf kostenfrei http://link.springer.com/article/10.1186/s13660-016-0962-y kostenfrei https://doaj.org/toc/1029-242X Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2016 1 7
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callnumber-first	Q - Science
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topic_title	QA1-939 Balls in generalizations of metric spaces ball partial b-metric space cone metric space
topic	misc QA1-939 misc ball misc partial b-metric space misc cone metric space misc Mathematics
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title	Balls in generalizations of metric spaces
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title_full	Balls in generalizations of metric spaces
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title_sort	balls in generalizations of metric spaces
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title_auth	Balls in generalizations of metric spaces
abstract	Abstract This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let ( X , p b ) $(X,p_{b})$ be a partial b-metric space in the sense of Mustafa et al. For the family △ of all p b $p_{b}$ -open balls in ( X , p b ) $(X,p_{b})$ , this paper proves that there are x , y ∈ B ∈ △ $x,y\in B\in\triangle$ such that B ′ ⊈ B $B'\nsubseteq B$ for all B ′ ∈ △ $B'\in\triangle$ , where B and B ′ $B'$ are with centers x and y, respectively. This result shows that △ is not a base of any topology on X, which shows that a proposition and a claim on partial b-metric spaces are not true. By some relations among ≪, <, and ≤ in cone metric spaces, this paper also constructs a cone metric space ( X , d ) $(X,d)$ and shows that { y ∈ X : d ( x , y ) ≪ ε } ‾ ≠ { y ∈ X : d ( x , y ) ≤ ε } $\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le\varepsilon\}$ in general, which corrects an error on cone metric spaces. However, it must be emphasized that these corrections do not affect the rest of the results in the relevant papers.
abstractGer	Abstract This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let ( X , p b ) $(X,p_{b})$ be a partial b-metric space in the sense of Mustafa et al. For the family △ of all p b $p_{b}$ -open balls in ( X , p b ) $(X,p_{b})$ , this paper proves that there are x , y ∈ B ∈ △ $x,y\in B\in\triangle$ such that B ′ ⊈ B $B'\nsubseteq B$ for all B ′ ∈ △ $B'\in\triangle$ , where B and B ′ $B'$ are with centers x and y, respectively. This result shows that △ is not a base of any topology on X, which shows that a proposition and a claim on partial b-metric spaces are not true. By some relations among ≪, <, and ≤ in cone metric spaces, this paper also constructs a cone metric space ( X , d ) $(X,d)$ and shows that { y ∈ X : d ( x , y ) ≪ ε } ‾ ≠ { y ∈ X : d ( x , y ) ≤ ε } $\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le\varepsilon\}$ in general, which corrects an error on cone metric spaces. However, it must be emphasized that these corrections do not affect the rest of the results in the relevant papers.
abstract_unstemmed	Abstract This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let ( X , p b ) $(X,p_{b})$ be a partial b-metric space in the sense of Mustafa et al. For the family △ of all p b $p_{b}$ -open balls in ( X , p b ) $(X,p_{b})$ , this paper proves that there are x , y ∈ B ∈ △ $x,y\in B\in\triangle$ such that B ′ ⊈ B $B'\nsubseteq B$ for all B ′ ∈ △ $B'\in\triangle$ , where B and B ′ $B'$ are with centers x and y, respectively. This result shows that △ is not a base of any topology on X, which shows that a proposition and a claim on partial b-metric spaces are not true. By some relations among ≪, <, and ≤ in cone metric spaces, this paper also constructs a cone metric space ( X , d ) $(X,d)$ and shows that { y ∈ X : d ( x , y ) ≪ ε } ‾ ≠ { y ∈ X : d ( x , y ) ≤ ε } $\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le\varepsilon\}$ in general, which corrects an error on cone metric spaces. However, it must be emphasized that these corrections do not affect the rest of the results in the relevant papers.
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title_short	Balls in generalizations of metric spaces
url	https://doi.org/10.1186/s13660-016-0962-y https://doaj.org/article/efc111cd81694827ab68106af9e532cf http://link.springer.com/article/10.1186/s13660-016-0962-y https://doaj.org/toc/1029-242X
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up_date	2024-07-03T17:07:14.470Z
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Balls in generalizations of metric spaces

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