What is a function that separates points of a manifold? [closed]
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In the context of differential geometry, what is a function that separates points of a manifold?
general-topology differential-geometry
edited Mar 15 at 11:57
YuiTo Cheng
2,1212837
asked Mar 15 at 9:46
mattiav27mattiav27
599
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closed as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, Cesareo Mar 16 at 9:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, dantopa, Parcly Taxel
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
In the context of differential geometry, what is a function that separates points of a manifold?
general-topology differential-geometry
edited Mar 15 at 11:57
YuiTo Cheng
2,1212837
asked Mar 15 at 9:46
mattiav27mattiav27
599
$endgroup$
closed as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, Cesareo Mar 16 at 9:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, dantopa, Parcly Taxel
If this question can be reworded to fit the rules in the help center, please edit the question.
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Can you give an example of what you mean with two points on a plane in $mathbb R^3 ? $
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– Narasimham
Mar 16 at 7:12
add a comment |
$begingroup$
In the context of differential geometry, what is a function that separates points of a manifold?
general-topology differential-geometry
edited Mar 15 at 11:57
YuiTo Cheng
2,1212837
asked Mar 15 at 9:46
mattiav27mattiav27
599
$endgroup$
In the context of differential geometry, what is a function that separates points of a manifold?
general-topology differential-geometry
general-topology differential-geometry
edited Mar 15 at 11:57
YuiTo Cheng
2,1212837
asked Mar 15 at 9:46
mattiav27mattiav27
599
edited Mar 15 at 11:57
YuiTo Cheng
2,1212837
asked Mar 15 at 9:46
mattiav27mattiav27
599
edited Mar 15 at 11:57
YuiTo Cheng
2,1212837
edited Mar 15 at 11:57
YuiTo Cheng
2,1212837
edited Mar 15 at 11:57
YuiTo Cheng
2,1212837
2,1212837
asked Mar 15 at 9:46
mattiav27mattiav27
599
asked Mar 15 at 9:46
mattiav27mattiav27
599
asked Mar 15 at 9:46
mattiav27mattiav27
599
599
closed as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, Cesareo Mar 16 at 9:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, dantopa, Parcly Taxel
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, Cesareo Mar 16 at 9:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, dantopa, Parcly Taxel
If this question can be reworded to fit the rules in the help center, please edit the question.
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Can you give an example of what you mean with two points on a plane in $mathbb R^3 ? $
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– Narasimham
Mar 16 at 7:12
add a comment |
$begingroup$
Can you give an example of what you mean with two points on a plane in $mathbb R^3 ? $
$endgroup$
– Narasimham
Mar 16 at 7:12
$begingroup$
Can you give an example of what you mean with two points on a plane in $mathbb R^3 ? $
$endgroup$
– Narasimham
Mar 16 at 7:12
$begingroup$
Can you give an example of what you mean with two points on a plane in $mathbb R^3 ? $
$endgroup$
– Narasimham
Mar 16 at 7:12
add a comment |
3 Answers
3
active
oldest
votes
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If you have a family $mathcal F$ of functions from a set $X$ into some other set, we say that it separates the points of $x$ if, for each $x_1,x_2in X$ with $x_1neq x_2$, there is some $finmathcal F$ such that $f(x_1)neq f(x_2)$.
So, if $mathcal F$ consists of a single function $f$, this is the same thing as asserting that $f$ is injective.
answered Mar 15 at 9:54
José Carlos SantosJosé Carlos Santos
170k23132238
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@ Jose: you should ad: $x_1 ne x_2.$
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– Fred
Mar 15 at 10:07
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@Fred I've edited my answer. Thank you.
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– José Carlos Santos
Mar 15 at 11:07
add a comment |
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A function that separates points is a (smooth) real-valued function which has different value at the different points. If we only have two points, it is possible (depending on your lecturer and / or textbook author) that such a function is required to have the value $0$ at one point and $1$ at the other.
answered Mar 15 at 9:53
ArthurArthur
119k7119202
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add a comment |
$begingroup$
Let $A$ and $B$ be sets and let $S$ be a set of functions $f:A to B.$ $S$ is said to separate the points of $A$, if for any $x,y in A$ with $x ne y$, there is $f in S$ such that $f(x) ne f(y)$.
answered Mar 15 at 9:54
FredFred
48.7k11849
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add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If you have a family $mathcal F$ of functions from a set $X$ into some other set, we say that it separates the points of $x$ if, for each $x_1,x_2in X$ with $x_1neq x_2$, there is some $finmathcal F$ such that $f(x_1)neq f(x_2)$.
So, if $mathcal F$ consists of a single function $f$, this is the same thing as asserting that $f$ is injective.
answered Mar 15 at 9:54
José Carlos SantosJosé Carlos Santos
170k23132238
$endgroup$
$begingroup$
@ Jose: you should ad: $x_1 ne x_2.$
$endgroup$
– Fred
Mar 15 at 10:07
$begingroup$
@Fred I've edited my answer. Thank you.
$endgroup$
– José Carlos Santos
Mar 15 at 11:07
add a comment |
$begingroup$
If you have a family $mathcal F$ of functions from a set $X$ into some other set, we say that it separates the points of $x$ if, for each $x_1,x_2in X$ with $x_1neq x_2$, there is some $finmathcal F$ such that $f(x_1)neq f(x_2)$.
So, if $mathcal F$ consists of a single function $f$, this is the same thing as asserting that $f$ is injective.
answered Mar 15 at 9:54
José Carlos SantosJosé Carlos Santos
170k23132238
$endgroup$
$begingroup$
@ Jose: you should ad: $x_1 ne x_2.$
$endgroup$
– Fred
Mar 15 at 10:07
$begingroup$
@Fred I've edited my answer. Thank you.
$endgroup$
– José Carlos Santos
Mar 15 at 11:07
add a comment |
$begingroup$
If you have a family $mathcal F$ of functions from a set $X$ into some other set, we say that it separates the points of $x$ if, for each $x_1,x_2in X$ with $x_1neq x_2$, there is some $finmathcal F$ such that $f(x_1)neq f(x_2)$.
So, if $mathcal F$ consists of a single function $f$, this is the same thing as asserting that $f$ is injective.
answered Mar 15 at 9:54
José Carlos SantosJosé Carlos Santos
170k23132238
$endgroup$
If you have a family $mathcal F$ of functions from a set $X$ into some other set, we say that it separates the points of $x$ if, for each $x_1,x_2in X$ with $x_1neq x_2$, there is some $finmathcal F$ such that $f(x_1)neq f(x_2)$.
So, if $mathcal F$ consists of a single function $f$, this is the same thing as asserting that $f$ is injective.
answered Mar 15 at 9:54
José Carlos SantosJosé Carlos Santos
170k23132238
edited Mar 15 at 11:07
answered Mar 15 at 9:54
José Carlos SantosJosé Carlos Santos
170k23132238
answered Mar 15 at 9:54
José Carlos SantosJosé Carlos Santos
170k23132238
answered Mar 15 at 9:54
José Carlos SantosJosé Carlos Santos
170k23132238
170k23132238
$begingroup$
@ Jose: you should ad: $x_1 ne x_2.$
$endgroup$
– Fred
Mar 15 at 10:07
$begingroup$
@Fred I've edited my answer. Thank you.
$endgroup$
– José Carlos Santos
Mar 15 at 11:07
add a comment |
$begingroup$
@ Jose: you should ad: $x_1 ne x_2.$
$endgroup$
– Fred
Mar 15 at 10:07
$begingroup$
@Fred I've edited my answer. Thank you.
$endgroup$
– José Carlos Santos
Mar 15 at 11:07
$begingroup$
@ Jose: you should ad: $x_1 ne x_2.$
$endgroup$
– Fred
Mar 15 at 10:07
$begingroup$
@ Jose: you should ad: $x_1 ne x_2.$
$endgroup$
– Fred
Mar 15 at 10:07
$begingroup$
@Fred I've edited my answer. Thank you.
$endgroup$
– José Carlos Santos
Mar 15 at 11:07
$begingroup$
@Fred I've edited my answer. Thank you.
$endgroup$
– José Carlos Santos
Mar 15 at 11:07
add a comment |
$begingroup$
A function that separates points is a (smooth) real-valued function which has different value at the different points. If we only have two points, it is possible (depending on your lecturer and / or textbook author) that such a function is required to have the value $0$ at one point and $1$ at the other.
answered Mar 15 at 9:53
ArthurArthur
119k7119202
$endgroup$
add a comment |
$begingroup$
A function that separates points is a (smooth) real-valued function which has different value at the different points. If we only have two points, it is possible (depending on your lecturer and / or textbook author) that such a function is required to have the value $0$ at one point and $1$ at the other.
answered Mar 15 at 9:53
ArthurArthur
119k7119202
$endgroup$
add a comment |
$begingroup$
A function that separates points is a (smooth) real-valued function which has different value at the different points. If we only have two points, it is possible (depending on your lecturer and / or textbook author) that such a function is required to have the value $0$ at one point and $1$ at the other.
answered Mar 15 at 9:53
ArthurArthur
119k7119202
$endgroup$
A function that separates points is a (smooth) real-valued function which has different value at the different points. If we only have two points, it is possible (depending on your lecturer and / or textbook author) that such a function is required to have the value $0$ at one point and $1$ at the other.
answered Mar 15 at 9:53
ArthurArthur
119k7119202
answered Mar 15 at 9:53
ArthurArthur
119k7119202
answered Mar 15 at 9:53
ArthurArthur
119k7119202
answered Mar 15 at 9:53
ArthurArthur
119k7119202
119k7119202
add a comment |
add a comment |
$begingroup$
Let $A$ and $B$ be sets and let $S$ be a set of functions $f:A to B.$ $S$ is said to separate the points of $A$, if for any $x,y in A$ with $x ne y$, there is $f in S$ such that $f(x) ne f(y)$.
answered Mar 15 at 9:54
FredFred
48.7k11849
$endgroup$
add a comment |
$begingroup$
Let $A$ and $B$ be sets and let $S$ be a set of functions $f:A to B.$ $S$ is said to separate the points of $A$, if for any $x,y in A$ with $x ne y$, there is $f in S$ such that $f(x) ne f(y)$.
answered Mar 15 at 9:54
FredFred
48.7k11849
$endgroup$
add a comment |
$begingroup$
Let $A$ and $B$ be sets and let $S$ be a set of functions $f:A to B.$ $S$ is said to separate the points of $A$, if for any $x,y in A$ with $x ne y$, there is $f in S$ such that $f(x) ne f(y)$.
answered Mar 15 at 9:54
FredFred
48.7k11849
$endgroup$
Let $A$ and $B$ be sets and let $S$ be a set of functions $f:A to B.$ $S$ is said to separate the points of $A$, if for any $x,y in A$ with $x ne y$, there is $f in S$ such that $f(x) ne f(y)$.
answered Mar 15 at 9:54
FredFred
48.7k11849
answered Mar 15 at 9:54
FredFred
48.7k11849
answered Mar 15 at 9:54
FredFred
48.7k11849
answered Mar 15 at 9:54
FredFred
48.7k11849
48.7k11849
add a comment |
add a comment |
$begingroup$
Can you give an example of what you mean with two points on a plane in $mathbb R^3 ? $
$endgroup$
– Narasimham
Mar 16 at 7:12