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

degree of differentiability of a manifold at a point$C(X)$ separates points?Why continuous paths implies smooth path on the manifold?What does it mean to “calculate in local coordinates” on a manifold?Closed ball not a manifold.What is the difference between intrinsic and extrinsic manifold?Showing an algebra separates pointsShow that the graph of $f$ is an immersed manifoldDerivation of a function on a manifoldHow can we show that $C_c^infty(mathbb R)$ strongly separates points?

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 yesterday

YuiTo Cheng

2,0532637

asked yesterday

mattiav27mattiav27

599

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put on hold as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, Cesareo 11 hours ago

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
  13 hours ago
  
  

  
  
  
  

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 yesterday

YuiTo Cheng

2,0532637

asked yesterday

mattiav27mattiav27

599

$endgroup$

put on hold as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, Cesareo 11 hours ago

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
  13 hours ago
  
  

  
  
  
  

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 yesterday

YuiTo Cheng

2,0532637

asked yesterday

mattiav27mattiav27

599

$endgroup$

share|cite|improve this question

edited yesterday

YuiTo Cheng

2,0532637

asked yesterday

mattiav27mattiav27

599

share|cite|improve this question

edited yesterday

YuiTo Cheng

2,0532637

asked yesterday

mattiav27mattiav27

599

edited yesterday

YuiTo Cheng

2,0532637

edited yesterday

YuiTo Cheng

2,0532637

asked yesterday

mattiav27mattiav27

599

asked yesterday

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 yesterday

answered yesterday

José Carlos SantosJosé Carlos Santos

168k22132236

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @ Jose: you should ad: $x_1 ne x_2.$
  $endgroup$
  – Fred
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Fred I've edited my answer. Thank you.
  $endgroup$
  – José Carlos Santos
  yesterday
  

  
  
  
  

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 yesterday

ArthurArthur

118k7118201

$endgroup$

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 yesterday

FredFred

48.3k1849

$endgroup$

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 yesterday

answered yesterday

José Carlos SantosJosé Carlos Santos

168k22132236

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @ Jose: you should ad: $x_1 ne x_2.$
  $endgroup$
  – Fred
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Fred I've edited my answer. Thank you.
  $endgroup$
  – José Carlos Santos
  yesterday
  

  
  
  
  

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 yesterday

answered yesterday

José Carlos SantosJosé Carlos Santos

168k22132236

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @ Jose: you should ad: $x_1 ne x_2.$
  $endgroup$
  – Fred
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Fred I've edited my answer. Thank you.
  $endgroup$
  – José Carlos Santos
  yesterday
  

  
  
  
  

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 yesterday

answered yesterday

José Carlos SantosJosé Carlos Santos

168k22132236

$endgroup$

share|cite|improve this answer

edited yesterday

answered yesterday

José Carlos SantosJosé Carlos Santos

168k22132236

answered yesterday

José Carlos SantosJosé Carlos Santos

168k22132236

answered yesterday

José Carlos SantosJosé Carlos Santos

168k22132236

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 yesterday

ArthurArthur

118k7118201

$endgroup$

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 yesterday

ArthurArthur

118k7118201

$endgroup$

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 yesterday

ArthurArthur

118k7118201

$endgroup$

share|cite|improve this answer

answered yesterday

ArthurArthur

118k7118201

answered yesterday

ArthurArthur

118k7118201

answered yesterday

ArthurArthur

118k7118201

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 yesterday

FredFred

48.3k1849

$endgroup$

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 yesterday

FredFred

48.3k1849

$endgroup$

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 yesterday

FredFred

48.3k1849

$endgroup$

share|cite|improve this answer

answered yesterday

FredFred

48.3k1849

answered yesterday

FredFred

48.3k1849

answered yesterday

FredFred

48.3k1849