There exists a point $y_0 in A $ with $d(y_0,x) = d(x,A)$ if choose the correct option












0














let A be a subset of $mathbb{R}^P$ and $ x in mathbb{R}^P$ denotes $d(x,A) = inf { d(x,y) : y in A}$.There exists a point $y_0 in A $ with $d(y_0,x) = d(x,A)$ if



choose the correct option



$a)$ $A$ is any closed non emepty subset of $mathbb{R}^P$



$b)$$ A$ is any non empty subset of $mathbb{R}^P$



$c)$$ A$ is any non empty compact subset of $mathbb{R}^P$



$d)$$ A$ is any non empty bounded subset of $mathbb{R}^P$



My attempt : i thinks option a) will correct because if A is closed then $x in bar A= A,$ that is $d(y_0,x) = d(x,A)=0$



I don't know that other option pliz help me






general-topology







share|cite|improve this question






















  • Why do you claim that $xinoverline A$?
    – José Carlos Santos
    Nov 30 at 23:51










  • because A is closed @ Jose carlos sir
    – jasmine
    Nov 30 at 23:56






  • 1




    That explains why you claim that $overline A=A$. But why $xinoverline A$?
    – José Carlos Santos
    Nov 30 at 23:58










  • @JoséCarlosSantos.....soory sir that is my misunderstanding
    – jasmine
    Nov 30 at 23:59
















0














let A be a subset of $mathbb{R}^P$ and $ x in mathbb{R}^P$ denotes $d(x,A) = inf { d(x,y) : y in A}$.There exists a point $y_0 in A $ with $d(y_0,x) = d(x,A)$ if



choose the correct option



$a)$ $A$ is any closed non emepty subset of $mathbb{R}^P$



$b)$$ A$ is any non empty subset of $mathbb{R}^P$



$c)$$ A$ is any non empty compact subset of $mathbb{R}^P$



$d)$$ A$ is any non empty bounded subset of $mathbb{R}^P$



My attempt : i thinks option a) will correct because if A is closed then $x in bar A= A,$ that is $d(y_0,x) = d(x,A)=0$



I don't know that other option pliz help me






general-topology







share|cite|improve this question






















  • Why do you claim that $xinoverline A$?
    – José Carlos Santos
    Nov 30 at 23:51










  • because A is closed @ Jose carlos sir
    – jasmine
    Nov 30 at 23:56






  • 1




    That explains why you claim that $overline A=A$. But why $xinoverline A$?
    – José Carlos Santos
    Nov 30 at 23:58










  • @JoséCarlosSantos.....soory sir that is my misunderstanding
    – jasmine
    Nov 30 at 23:59














0












0








0







let A be a subset of $mathbb{R}^P$ and $ x in mathbb{R}^P$ denotes $d(x,A) = inf { d(x,y) : y in A}$.There exists a point $y_0 in A $ with $d(y_0,x) = d(x,A)$ if



choose the correct option



$a)$ $A$ is any closed non emepty subset of $mathbb{R}^P$



$b)$$ A$ is any non empty subset of $mathbb{R}^P$



$c)$$ A$ is any non empty compact subset of $mathbb{R}^P$



$d)$$ A$ is any non empty bounded subset of $mathbb{R}^P$



My attempt : i thinks option a) will correct because if A is closed then $x in bar A= A,$ that is $d(y_0,x) = d(x,A)=0$



I don't know that other option pliz help me






general-topology







share|cite|improve this question













let A be a subset of $mathbb{R}^P$ and $ x in mathbb{R}^P$ denotes $d(x,A) = inf { d(x,y) : y in A}$.There exists a point $y_0 in A $ with $d(y_0,x) = d(x,A)$ if



choose the correct option



$a)$ $A$ is any closed non emepty subset of $mathbb{R}^P$



$b)$$ A$ is any non empty subset of $mathbb{R}^P$



$c)$$ A$ is any non empty compact subset of $mathbb{R}^P$



$d)$$ A$ is any non empty bounded subset of $mathbb{R}^P$



My attempt : i thinks option a) will correct because if A is closed then $x in bar A= A,$ that is $d(y_0,x) = d(x,A)=0$



I don't know that other option pliz help me






general-topology




general-topology






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 30 at 23:46









jasmine

1,520416




1,520416












  • Why do you claim that $xinoverline A$?
    – José Carlos Santos
    Nov 30 at 23:51










  • because A is closed @ Jose carlos sir
    – jasmine
    Nov 30 at 23:56






  • 1




    That explains why you claim that $overline A=A$. But why $xinoverline A$?
    – José Carlos Santos
    Nov 30 at 23:58










  • @JoséCarlosSantos.....soory sir that is my misunderstanding
    – jasmine
    Nov 30 at 23:59


















  • Why do you claim that $xinoverline A$?
    – José Carlos Santos
    Nov 30 at 23:51










  • because A is closed @ Jose carlos sir
    – jasmine
    Nov 30 at 23:56






  • 1




    That explains why you claim that $overline A=A$. But why $xinoverline A$?
    – José Carlos Santos
    Nov 30 at 23:58










  • @JoséCarlosSantos.....soory sir that is my misunderstanding
    – jasmine
    Nov 30 at 23:59
















Why do you claim that $xinoverline A$?
– José Carlos Santos
Nov 30 at 23:51




Why do you claim that $xinoverline A$?
– José Carlos Santos
Nov 30 at 23:51












because A is closed @ Jose carlos sir
– jasmine
Nov 30 at 23:56




because A is closed @ Jose carlos sir
– jasmine
Nov 30 at 23:56




1




1




That explains why you claim that $overline A=A$. But why $xinoverline A$?
– José Carlos Santos
Nov 30 at 23:58




That explains why you claim that $overline A=A$. But why $xinoverline A$?
– José Carlos Santos
Nov 30 at 23:58












@JoséCarlosSantos.....soory sir that is my misunderstanding
– jasmine
Nov 30 at 23:59




@JoséCarlosSantos.....soory sir that is my misunderstanding
– jasmine
Nov 30 at 23:59










2 Answers
2






active

oldest

votes


















2














a) It is correct. Take $ain A$ and consider the closed ball $overline{B_{d(x,a)}(x)}$. Then $ainoverline{B_{d(x,a)}(x)}cap A$ and, since $overline{B_{d(x,a)}(x)}cap A$ is closed and bounded, it is compact. So, there is a $a'inoverline{B_{d(x,a)}(x)}cap A$ such that$$d(x,a')=min{d(x,c),|,cinoverline{B_{d(x,a)}(x)}cap A}.$$And if $cin Asetminusleft(overline{B_{d(x,a)}(x)}cap Aright)$, then $d(x,c)>d(x,a)geqslant d(x,a')$. So, $d(x,a')=d(x,A)$.



b) It is false. take $x=(0,0)$ and $A=(0,1)times{0}$.



c) It is correct, since a) holds.



d) It is false. Consider the same example as inmy answer to b).






share|cite|improve this answer





















  • carlos sir what is $y_0$ in option b) how can i contradict im getting$ d(y_0,x) = d(x,A) = 0 = d( 0, (0,1))$?
    – jasmine
    Dec 1 at 0:48






  • 1




    Option b) is false, which means that there is no $y_0$. In my example, $d(x,A)=1$, but there is no $y_0in A$ such that $d(x,y_0)=1$.
    – José Carlos Santos
    Dec 1 at 7:33










  • Strictly speaking, the argument for a) depends on the correctness of c) and vie versa.
    – random
    Dec 1 at 12:02



















2














a) and c) are true and b) and d) are false. For a) choose ${x_k} subset A$ such that $d(x,x_k)$ converges to $d(x,A)$. Then ${x_k}$ is bouunded, so it has a convergent subsequence. If the limit of the subsequence is $y$ the $ y in A$ and $d(x,y)=d(x,A)$. C) follows from a). For b) and d) Take $x=0$ and $A=(0,1)$ in $mathbb R$. Thanks to Jose carlos Santos for pointing out an error in my earlier answer.






share|cite|improve this answer























  • thanks u @Kavi sir ,,,i need detail
    – jasmine
    Nov 30 at 23:59






  • 1




    This is not correct, since option a) is also true.
    – José Carlos Santos
    Dec 1 at 0:07






  • 1




    @JoséCarlosSantos Thanks. I have edited my answer using your comment. My answer is a bit different so I though of retaining it. I hope you have no objection to this.
    – Kavi Rama Murthy
    Dec 1 at 0:42






  • 1




    Of course I have no objection.
    – José Carlos Santos
    Dec 1 at 7:31











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2 Answers
2






active

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2 Answers
2






active

oldest

votes









active

oldest

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active

oldest

votes









2














a) It is correct. Take $ain A$ and consider the closed ball $overline{B_{d(x,a)}(x)}$. Then $ainoverline{B_{d(x,a)}(x)}cap A$ and, since $overline{B_{d(x,a)}(x)}cap A$ is closed and bounded, it is compact. So, there is a $a'inoverline{B_{d(x,a)}(x)}cap A$ such that$$d(x,a')=min{d(x,c),|,cinoverline{B_{d(x,a)}(x)}cap A}.$$And if $cin Asetminusleft(overline{B_{d(x,a)}(x)}cap Aright)$, then $d(x,c)>d(x,a)geqslant d(x,a')$. So, $d(x,a')=d(x,A)$.



b) It is false. take $x=(0,0)$ and $A=(0,1)times{0}$.



c) It is correct, since a) holds.



d) It is false. Consider the same example as inmy answer to b).






share|cite|improve this answer





















  • carlos sir what is $y_0$ in option b) how can i contradict im getting$ d(y_0,x) = d(x,A) = 0 = d( 0, (0,1))$?
    – jasmine
    Dec 1 at 0:48






  • 1




    Option b) is false, which means that there is no $y_0$. In my example, $d(x,A)=1$, but there is no $y_0in A$ such that $d(x,y_0)=1$.
    – José Carlos Santos
    Dec 1 at 7:33










  • Strictly speaking, the argument for a) depends on the correctness of c) and vie versa.
    – random
    Dec 1 at 12:02
















2














a) It is correct. Take $ain A$ and consider the closed ball $overline{B_{d(x,a)}(x)}$. Then $ainoverline{B_{d(x,a)}(x)}cap A$ and, since $overline{B_{d(x,a)}(x)}cap A$ is closed and bounded, it is compact. So, there is a $a'inoverline{B_{d(x,a)}(x)}cap A$ such that$$d(x,a')=min{d(x,c),|,cinoverline{B_{d(x,a)}(x)}cap A}.$$And if $cin Asetminusleft(overline{B_{d(x,a)}(x)}cap Aright)$, then $d(x,c)>d(x,a)geqslant d(x,a')$. So, $d(x,a')=d(x,A)$.



b) It is false. take $x=(0,0)$ and $A=(0,1)times{0}$.



c) It is correct, since a) holds.



d) It is false. Consider the same example as inmy answer to b).






share|cite|improve this answer





















  • carlos sir what is $y_0$ in option b) how can i contradict im getting$ d(y_0,x) = d(x,A) = 0 = d( 0, (0,1))$?
    – jasmine
    Dec 1 at 0:48






  • 1




    Option b) is false, which means that there is no $y_0$. In my example, $d(x,A)=1$, but there is no $y_0in A$ such that $d(x,y_0)=1$.
    – José Carlos Santos
    Dec 1 at 7:33










  • Strictly speaking, the argument for a) depends on the correctness of c) and vie versa.
    – random
    Dec 1 at 12:02














2












2








2






a) It is correct. Take $ain A$ and consider the closed ball $overline{B_{d(x,a)}(x)}$. Then $ainoverline{B_{d(x,a)}(x)}cap A$ and, since $overline{B_{d(x,a)}(x)}cap A$ is closed and bounded, it is compact. So, there is a $a'inoverline{B_{d(x,a)}(x)}cap A$ such that$$d(x,a')=min{d(x,c),|,cinoverline{B_{d(x,a)}(x)}cap A}.$$And if $cin Asetminusleft(overline{B_{d(x,a)}(x)}cap Aright)$, then $d(x,c)>d(x,a)geqslant d(x,a')$. So, $d(x,a')=d(x,A)$.



b) It is false. take $x=(0,0)$ and $A=(0,1)times{0}$.



c) It is correct, since a) holds.



d) It is false. Consider the same example as inmy answer to b).






share|cite|improve this answer












a) It is correct. Take $ain A$ and consider the closed ball $overline{B_{d(x,a)}(x)}$. Then $ainoverline{B_{d(x,a)}(x)}cap A$ and, since $overline{B_{d(x,a)}(x)}cap A$ is closed and bounded, it is compact. So, there is a $a'inoverline{B_{d(x,a)}(x)}cap A$ such that$$d(x,a')=min{d(x,c),|,cinoverline{B_{d(x,a)}(x)}cap A}.$$And if $cin Asetminusleft(overline{B_{d(x,a)}(x)}cap Aright)$, then $d(x,c)>d(x,a)geqslant d(x,a')$. So, $d(x,a')=d(x,A)$.



b) It is false. take $x=(0,0)$ and $A=(0,1)times{0}$.



c) It is correct, since a) holds.



d) It is false. Consider the same example as inmy answer to b).







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 1 at 0:15









José Carlos Santos

148k22117219




148k22117219












  • carlos sir what is $y_0$ in option b) how can i contradict im getting$ d(y_0,x) = d(x,A) = 0 = d( 0, (0,1))$?
    – jasmine
    Dec 1 at 0:48






  • 1




    Option b) is false, which means that there is no $y_0$. In my example, $d(x,A)=1$, but there is no $y_0in A$ such that $d(x,y_0)=1$.
    – José Carlos Santos
    Dec 1 at 7:33










  • Strictly speaking, the argument for a) depends on the correctness of c) and vie versa.
    – random
    Dec 1 at 12:02


















  • carlos sir what is $y_0$ in option b) how can i contradict im getting$ d(y_0,x) = d(x,A) = 0 = d( 0, (0,1))$?
    – jasmine
    Dec 1 at 0:48






  • 1




    Option b) is false, which means that there is no $y_0$. In my example, $d(x,A)=1$, but there is no $y_0in A$ such that $d(x,y_0)=1$.
    – José Carlos Santos
    Dec 1 at 7:33










  • Strictly speaking, the argument for a) depends on the correctness of c) and vie versa.
    – random
    Dec 1 at 12:02
















carlos sir what is $y_0$ in option b) how can i contradict im getting$ d(y_0,x) = d(x,A) = 0 = d( 0, (0,1))$?
– jasmine
Dec 1 at 0:48




carlos sir what is $y_0$ in option b) how can i contradict im getting$ d(y_0,x) = d(x,A) = 0 = d( 0, (0,1))$?
– jasmine
Dec 1 at 0:48




1




1




Option b) is false, which means that there is no $y_0$. In my example, $d(x,A)=1$, but there is no $y_0in A$ such that $d(x,y_0)=1$.
– José Carlos Santos
Dec 1 at 7:33




Option b) is false, which means that there is no $y_0$. In my example, $d(x,A)=1$, but there is no $y_0in A$ such that $d(x,y_0)=1$.
– José Carlos Santos
Dec 1 at 7:33












Strictly speaking, the argument for a) depends on the correctness of c) and vie versa.
– random
Dec 1 at 12:02




Strictly speaking, the argument for a) depends on the correctness of c) and vie versa.
– random
Dec 1 at 12:02











2














a) and c) are true and b) and d) are false. For a) choose ${x_k} subset A$ such that $d(x,x_k)$ converges to $d(x,A)$. Then ${x_k}$ is bouunded, so it has a convergent subsequence. If the limit of the subsequence is $y$ the $ y in A$ and $d(x,y)=d(x,A)$. C) follows from a). For b) and d) Take $x=0$ and $A=(0,1)$ in $mathbb R$. Thanks to Jose carlos Santos for pointing out an error in my earlier answer.






share|cite|improve this answer























  • thanks u @Kavi sir ,,,i need detail
    – jasmine
    Nov 30 at 23:59






  • 1




    This is not correct, since option a) is also true.
    – José Carlos Santos
    Dec 1 at 0:07






  • 1




    @JoséCarlosSantos Thanks. I have edited my answer using your comment. My answer is a bit different so I though of retaining it. I hope you have no objection to this.
    – Kavi Rama Murthy
    Dec 1 at 0:42






  • 1




    Of course I have no objection.
    – José Carlos Santos
    Dec 1 at 7:31
















2














a) and c) are true and b) and d) are false. For a) choose ${x_k} subset A$ such that $d(x,x_k)$ converges to $d(x,A)$. Then ${x_k}$ is bouunded, so it has a convergent subsequence. If the limit of the subsequence is $y$ the $ y in A$ and $d(x,y)=d(x,A)$. C) follows from a). For b) and d) Take $x=0$ and $A=(0,1)$ in $mathbb R$. Thanks to Jose carlos Santos for pointing out an error in my earlier answer.






share|cite|improve this answer























  • thanks u @Kavi sir ,,,i need detail
    – jasmine
    Nov 30 at 23:59






  • 1




    This is not correct, since option a) is also true.
    – José Carlos Santos
    Dec 1 at 0:07






  • 1




    @JoséCarlosSantos Thanks. I have edited my answer using your comment. My answer is a bit different so I though of retaining it. I hope you have no objection to this.
    – Kavi Rama Murthy
    Dec 1 at 0:42






  • 1




    Of course I have no objection.
    – José Carlos Santos
    Dec 1 at 7:31














2












2








2






a) and c) are true and b) and d) are false. For a) choose ${x_k} subset A$ such that $d(x,x_k)$ converges to $d(x,A)$. Then ${x_k}$ is bouunded, so it has a convergent subsequence. If the limit of the subsequence is $y$ the $ y in A$ and $d(x,y)=d(x,A)$. C) follows from a). For b) and d) Take $x=0$ and $A=(0,1)$ in $mathbb R$. Thanks to Jose carlos Santos for pointing out an error in my earlier answer.






share|cite|improve this answer














a) and c) are true and b) and d) are false. For a) choose ${x_k} subset A$ such that $d(x,x_k)$ converges to $d(x,A)$. Then ${x_k}$ is bouunded, so it has a convergent subsequence. If the limit of the subsequence is $y$ the $ y in A$ and $d(x,y)=d(x,A)$. C) follows from a). For b) and d) Take $x=0$ and $A=(0,1)$ in $mathbb R$. Thanks to Jose carlos Santos for pointing out an error in my earlier answer.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 1 at 1:51

























answered Nov 30 at 23:57









Kavi Rama Murthy

49.1k31854




49.1k31854












  • thanks u @Kavi sir ,,,i need detail
    – jasmine
    Nov 30 at 23:59






  • 1




    This is not correct, since option a) is also true.
    – José Carlos Santos
    Dec 1 at 0:07






  • 1




    @JoséCarlosSantos Thanks. I have edited my answer using your comment. My answer is a bit different so I though of retaining it. I hope you have no objection to this.
    – Kavi Rama Murthy
    Dec 1 at 0:42






  • 1




    Of course I have no objection.
    – José Carlos Santos
    Dec 1 at 7:31


















  • thanks u @Kavi sir ,,,i need detail
    – jasmine
    Nov 30 at 23:59






  • 1




    This is not correct, since option a) is also true.
    – José Carlos Santos
    Dec 1 at 0:07






  • 1




    @JoséCarlosSantos Thanks. I have edited my answer using your comment. My answer is a bit different so I though of retaining it. I hope you have no objection to this.
    – Kavi Rama Murthy
    Dec 1 at 0:42






  • 1




    Of course I have no objection.
    – José Carlos Santos
    Dec 1 at 7:31
















thanks u @Kavi sir ,,,i need detail
– jasmine
Nov 30 at 23:59




thanks u @Kavi sir ,,,i need detail
– jasmine
Nov 30 at 23:59




1




1




This is not correct, since option a) is also true.
– José Carlos Santos
Dec 1 at 0:07




This is not correct, since option a) is also true.
– José Carlos Santos
Dec 1 at 0:07




1




1




@JoséCarlosSantos Thanks. I have edited my answer using your comment. My answer is a bit different so I though of retaining it. I hope you have no objection to this.
– Kavi Rama Murthy
Dec 1 at 0:42




@JoséCarlosSantos Thanks. I have edited my answer using your comment. My answer is a bit different so I though of retaining it. I hope you have no objection to this.
– Kavi Rama Murthy
Dec 1 at 0:42




1




1




Of course I have no objection.
– José Carlos Santos
Dec 1 at 7:31




Of course I have no objection.
– José Carlos Santos
Dec 1 at 7:31


















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Berounka

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Berounka.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Caractéristiques Longueur 139  km Bassin 8 861  km 2 Bassin collecteur Elbe Débit moyen ? Cours Source Confluence de la Mže et de la Radbuza · Localisation Pilsen · Altitude 305  m · Coordonnées 49° 45′ 13″ N, 13° 23′ 24″ E Confluence Vltava · Localisation Prague · Altitude 187  m · Coordonnées 49° 59′ 43″ N, 14° 24′ 04″ E Géographie Pays traversés   République tchèque modifier   La Berounka est une rivière de la République tchèque. Elle porte le nom de Mže depuis sa source en Allemagne près de la frontière jusqu'à sa confluence avec la Radbuza à Plzeň. Elle continue ensuite sous le nom de Berounka jusqu'à sa confluence avec la Vltava, près de Prague. La rivière est une destination prisée pour les amateurs de canoë, qui appréc...
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Sphinx de Gizeh

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Pour les articles homonymes, voir Sphinx. Sphinx de Gizeh.mw-parser-output .entete.egypte-antique{background-image:url("//upload.wikimedia.org/wikipedia/commons/c/cc/Picto_infobox_ancient_Egypt.png")} Site d'Égypte antique Le visage du sphinx de Gizeh. Noms en arabe أبو الهول , Abou al-Hôl Localisation Région Basse-Égypte Coordonnées 29° 58′ 31″ nord, 31° 08′ 16″ est Géolocalisation sur la carte : Égypte Sphinx de Gizeh modifier   Le sphinx de Gizeh est la statue thérianthrope qui se dresse devant les grandes pyramides du plateau de Gizeh, en Basse-Égypte. Sculpture monumentale monolithique la plus grande du monde [ 1 ] avec 73,5 mètres de longueur, 14 mètres de largeur et 20,22 mètres de hauteur, elle représente un sphinx couchant. Réalisée vers -2500, elle serait attribuée à Djédefrê, l'un des pharaons de la IV e  dynastie, représentant son père le pharaon Khéops. Sommaire 1 Étymologie 2 Desc...
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Different font size/position of beamer's navigation symbols template's content depending on regular/plain...

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3 Let's suppose I'd like to divert the navigation symbols template (e.g. in order replace them by the frame numbers, see this bug report ;) I would be facing troubles regarding the font size/position, depending on regular/plain frames, as shown by the following MCE. How could I get rid of these troubles? documentclass{beamer} usetheme{PaloAlto} setbeamertemplate{navigation symbols}{insertframenumber/inserttotalframenumber} begin{document} begin{frame} frametitle{A regular frame} end{frame} begin{frame}[plain] frametitle{A plain frame} end{frame} end{document} beamer header-footer share | improve this question asked Dec 10 '18 a...
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