How can I define a neighborhood (not only $delta$ neighborhood) in $mathbb R$. [duplicate]
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This question already has an answer here:
Definition of neighborhood and open set in topology
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How can I define a neighborhood (not only $delta$ neighborhood) of a point in $mathbb R$.
without using metric concept.
According to rudin's definition which must be a open set.
real-analysis
asked Jan 12 at 11:45
Supriyo BanerjeeSupriyo Banerjee
1036
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marked as duplicate by Saad, Adrian Keister, Michael Hoppe, José Carlos Santos real-analysis
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Jan 12 at 16:19
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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This question already has an answer here:
Definition of neighborhood and open set in topology
8 answers
How can I define a neighborhood (not only $delta$ neighborhood) of a point in $mathbb R$.
without using metric concept.
According to rudin's definition which must be a open set.
real-analysis
asked Jan 12 at 11:45
Supriyo BanerjeeSupriyo Banerjee
1036
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marked as duplicate by Saad, Adrian Keister, Michael Hoppe, José Carlos Santos real-analysis
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Jan 12 at 16:19
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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If $mathcal T $ is a topology on $Bbb R $, a neighborhood of a point $a$ is any open set $U$ (that is $Uin mathcal T $) containing $a$.
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– Thomas Shelby
Jan 12 at 11:54
add a comment |
$begingroup$
This question already has an answer here:
Definition of neighborhood and open set in topology
8 answers
How can I define a neighborhood (not only $delta$ neighborhood) of a point in $mathbb R$.
without using metric concept.
According to rudin's definition which must be a open set.
real-analysis
asked Jan 12 at 11:45
Supriyo BanerjeeSupriyo Banerjee
1036
$endgroup$
This question already has an answer here:
Definition of neighborhood and open set in topology
8 answers
How can I define a neighborhood (not only $delta$ neighborhood) of a point in $mathbb R$.
without using metric concept.
According to rudin's definition which must be a open set.
This question already has an answer here:
Definition of neighborhood and open set in topology
8 answers
real-analysis
real-analysis
asked Jan 12 at 11:45
Supriyo BanerjeeSupriyo Banerjee
1036
asked Jan 12 at 11:45
Supriyo BanerjeeSupriyo Banerjee
1036
asked Jan 12 at 11:45
Supriyo BanerjeeSupriyo Banerjee
1036
asked Jan 12 at 11:45
Supriyo BanerjeeSupriyo Banerjee
1036
asked Jan 12 at 11:45
Supriyo BanerjeeSupriyo Banerjee
1036
1036
marked as duplicate by Saad, Adrian Keister, Michael Hoppe, José Carlos Santos real-analysis
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Jan 12 at 16:19
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Jan 12 at 16:19
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
If $mathcal T $ is a topology on $Bbb R $, a neighborhood of a point $a$ is any open set $U$ (that is $Uin mathcal T $) containing $a$.
$endgroup$
– Thomas Shelby
Jan 12 at 11:54
add a comment |
$begingroup$
If $mathcal T $ is a topology on $Bbb R $, a neighborhood of a point $a$ is any open set $U$ (that is $Uin mathcal T $) containing $a$.
$endgroup$
– Thomas Shelby
Jan 12 at 11:54
$begingroup$
If $mathcal T $ is a topology on $Bbb R $, a neighborhood of a point $a$ is any open set $U$ (that is $Uin mathcal T $) containing $a$.
$endgroup$
– Thomas Shelby
Jan 12 at 11:54
$begingroup$
If $mathcal T $ is a topology on $Bbb R $, a neighborhood of a point $a$ is any open set $U$ (that is $Uin mathcal T $) containing $a$.
$endgroup$
– Thomas Shelby
Jan 12 at 11:54
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You need some concept. If you don't want metric concept, then you at the very least need topology. And in topology, rather than a metric, all you have available to work with is the collection of all subsets of $Bbb R$ which we call "open".
In this context, a neighborhood of $xinBbb R$ is simply any such open set which contains $x$.
answered Jan 12 at 11:58
ArthurArthur
118k7117200
$endgroup$
2
$begingroup$
A small clarification: in some books any set which contains an open set containing $x$ is called a neighborhood of $x$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 12:06
$begingroup$
@KaviRamaMurthy exactly.In many undergrad books it is written.
$endgroup$
– Supriyo Banerjee
Jan 12 at 12:11
$begingroup$
@KaviRamaMurthy You're right. With that convention, an open neighborhood is an open set containing the point, and a neighborhood is any set which contains an open neighborhood.
$endgroup$
– Arthur
Jan 12 at 14:05
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You need some concept. If you don't want metric concept, then you at the very least need topology. And in topology, rather than a metric, all you have available to work with is the collection of all subsets of $Bbb R$ which we call "open".
In this context, a neighborhood of $xinBbb R$ is simply any such open set which contains $x$.
answered Jan 12 at 11:58
ArthurArthur
118k7117200
$endgroup$
2
$begingroup$
A small clarification: in some books any set which contains an open set containing $x$ is called a neighborhood of $x$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 12:06
$begingroup$
@KaviRamaMurthy exactly.In many undergrad books it is written.
$endgroup$
– Supriyo Banerjee
Jan 12 at 12:11
$begingroup$
@KaviRamaMurthy You're right. With that convention, an open neighborhood is an open set containing the point, and a neighborhood is any set which contains an open neighborhood.
$endgroup$
– Arthur
Jan 12 at 14:05
add a comment |
$begingroup$
You need some concept. If you don't want metric concept, then you at the very least need topology. And in topology, rather than a metric, all you have available to work with is the collection of all subsets of $Bbb R$ which we call "open".
In this context, a neighborhood of $xinBbb R$ is simply any such open set which contains $x$.
answered Jan 12 at 11:58
ArthurArthur
118k7117200
$endgroup$
2
$begingroup$
A small clarification: in some books any set which contains an open set containing $x$ is called a neighborhood of $x$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 12:06
$begingroup$
@KaviRamaMurthy exactly.In many undergrad books it is written.
$endgroup$
– Supriyo Banerjee
Jan 12 at 12:11
$begingroup$
@KaviRamaMurthy You're right. With that convention, an open neighborhood is an open set containing the point, and a neighborhood is any set which contains an open neighborhood.
$endgroup$
– Arthur
Jan 12 at 14:05
add a comment |
$begingroup$
You need some concept. If you don't want metric concept, then you at the very least need topology. And in topology, rather than a metric, all you have available to work with is the collection of all subsets of $Bbb R$ which we call "open".
In this context, a neighborhood of $xinBbb R$ is simply any such open set which contains $x$.
answered Jan 12 at 11:58
ArthurArthur
118k7117200
$endgroup$
You need some concept. If you don't want metric concept, then you at the very least need topology. And in topology, rather than a metric, all you have available to work with is the collection of all subsets of $Bbb R$ which we call "open".
In this context, a neighborhood of $xinBbb R$ is simply any such open set which contains $x$.
answered Jan 12 at 11:58
ArthurArthur
118k7117200
answered Jan 12 at 11:58
ArthurArthur
118k7117200
answered Jan 12 at 11:58
ArthurArthur
118k7117200
answered Jan 12 at 11:58
ArthurArthur
118k7117200
118k7117200
2
$begingroup$
A small clarification: in some books any set which contains an open set containing $x$ is called a neighborhood of $x$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 12:06
$begingroup$
@KaviRamaMurthy exactly.In many undergrad books it is written.
$endgroup$
– Supriyo Banerjee
Jan 12 at 12:11
$begingroup$
@KaviRamaMurthy You're right. With that convention, an open neighborhood is an open set containing the point, and a neighborhood is any set which contains an open neighborhood.
$endgroup$
– Arthur
Jan 12 at 14:05
add a comment |
2
$begingroup$
A small clarification: in some books any set which contains an open set containing $x$ is called a neighborhood of $x$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 12:06
$begingroup$
@KaviRamaMurthy exactly.In many undergrad books it is written.
$endgroup$
– Supriyo Banerjee
Jan 12 at 12:11
$begingroup$
@KaviRamaMurthy You're right. With that convention, an open neighborhood is an open set containing the point, and a neighborhood is any set which contains an open neighborhood.
$endgroup$
– Arthur
Jan 12 at 14:05
2
2
$begingroup$
A small clarification: in some books any set which contains an open set containing $x$ is called a neighborhood of $x$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 12:06
$begingroup$
A small clarification: in some books any set which contains an open set containing $x$ is called a neighborhood of $x$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 12:06
$begingroup$
@KaviRamaMurthy exactly.In many undergrad books it is written.
$endgroup$
– Supriyo Banerjee
Jan 12 at 12:11
$begingroup$
@KaviRamaMurthy exactly.In many undergrad books it is written.
$endgroup$
– Supriyo Banerjee
Jan 12 at 12:11
$begingroup$
@KaviRamaMurthy You're right. With that convention, an open neighborhood is an open set containing the point, and a neighborhood is any set which contains an open neighborhood.
$endgroup$
– Arthur
Jan 12 at 14:05
$begingroup$
@KaviRamaMurthy You're right. With that convention, an open neighborhood is an open set containing the point, and a neighborhood is any set which contains an open neighborhood.
$endgroup$
– Arthur
Jan 12 at 14:05
add a comment |
$begingroup$
If $mathcal T $ is a topology on $Bbb R $, a neighborhood of a point $a$ is any open set $U$ (that is $Uin mathcal T $) containing $a$.
$endgroup$
– Thomas Shelby
Jan 12 at 11:54