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What is a function that separates points of a manifold? [closed]

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2

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In the context of differential geometry, what is a function that separates points of a manifold?

general-topology differential-geometry

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edited Mar 15 at 11:57

YuiTo Cheng

2,1362837

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.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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 | 

2

$begingroup$

In the context of differential geometry, what is a function that separates points of a manifold?

general-topology differential-geometry

share|cite|improve this question

edited Mar 15 at 11:57

YuiTo Cheng

2,1362837

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.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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 | 

$begingroup$

In the context of differential geometry, what is a function that separates points of a manifold?

general-topology differential-geometry

share|cite|improve this question

edited Mar 15 at 11:57

YuiTo Cheng

2,1362837

asked Mar 15 at 9:46

mattiav27mattiav27

599

$endgroup$

share|cite|improve this question

edited Mar 15 at 11:57

YuiTo Cheng

2,1362837

asked Mar 15 at 9:46

mattiav27mattiav27

599

share|cite|improve this question

edited Mar 15 at 11:57

YuiTo Cheng

2,1362837

asked Mar 15 at 9:46

mattiav27mattiav27

599

edited Mar 15 at 11:57

YuiTo Cheng

2,1362837

edited Mar 15 at 11:57

YuiTo Cheng

2,1362837

asked Mar 15 at 9:46

mattiav27mattiav27

599

asked Mar 15 at 9:46

mattiav27mattiav27

599

3 Answers
3

active

oldest

votes

6

$begingroup$

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.

share|cite|improve this answer

edited Mar 15 at 11:07

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 | 

6

$begingroup$

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.

share|cite|improve this answer

answered Mar 15 at 9:53

ArthurArthur

120k7120203

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add a comment | 

4

$begingroup$

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 $f(x) ne f(y)$.

share|cite|improve this answer

answered Mar 15 at 9:54

FredFred

48.8k11849

$endgroup$

add a comment | 

6

$begingroup$

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.

share|cite|improve this answer

edited Mar 15 at 11:07

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 | 

6

$begingroup$

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.

share|cite|improve this answer

edited Mar 15 at 11:07

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.

share|cite|improve this answer

edited Mar 15 at 11:07

answered Mar 15 at 9:54

José Carlos SantosJosé Carlos Santos

170k23132238

$endgroup$

share|cite|improve this answer

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

6

$begingroup$

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.

share|cite|improve this answer

answered Mar 15 at 9:53

ArthurArthur

120k7120203

$endgroup$

add a comment | 

6

$begingroup$

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.

share|cite|improve this answer

answered Mar 15 at 9:53

ArthurArthur

120k7120203

$endgroup$

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.

share|cite|improve this answer

answered Mar 15 at 9:53

ArthurArthur

120k7120203

$endgroup$

share|cite|improve this answer

answered Mar 15 at 9:53

ArthurArthur

120k7120203

answered Mar 15 at 9:53

ArthurArthur

120k7120203

answered Mar 15 at 9:53

ArthurArthur

120k7120203

4

$begingroup$

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 $f(x) ne f(y)$.

share|cite|improve this answer

answered Mar 15 at 9:54

FredFred

48.8k11849

$endgroup$

add a comment | 

4

$begingroup$

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 $f(x) ne f(y)$.

share|cite|improve this answer

answered Mar 15 at 9:54

FredFred

48.8k11849

$endgroup$

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 $f(x) ne f(y)$.

share|cite|improve this answer

answered Mar 15 at 9:54

FredFred

48.8k11849

$endgroup$

share|cite|improve this answer

answered Mar 15 at 9:54

FredFred

48.8k11849

answered Mar 15 at 9:54

FredFred

48.8k11849

answered Mar 15 at 9:54

FredFred

48.8k11849