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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












$begingroup$


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











$endgroup$



put on hold as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, Cesareo 12 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
    14 hours ago
















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











$endgroup$



put on hold as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, Cesareo 12 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
    14 hours ago














2












2








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











$endgroup$




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






general-topology differential-geometry




general-topology differential-geometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday









YuiTo Cheng

2,0532637




2,0532637










asked yesterday









mattiav27mattiav27

599




599




put on hold as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, Cesareo 12 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.







put on hold as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, Cesareo 12 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
    14 hours ago

















  • $begingroup$
    Can you give an example of what you mean with two points on a plane in $mathbb R^3 ? $
    $endgroup$
    – Narasimham
    14 hours ago
















$begingroup$
Can you give an example of what you mean with two points on a plane in $mathbb R^3 ? $
$endgroup$
– Narasimham
14 hours ago





$begingroup$
Can you give an example of what you mean with two points on a plane in $mathbb R^3 ? $
$endgroup$
– Narasimham
14 hours ago











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











$endgroup$












  • $begingroup$
    @ 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


















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









$endgroup$




















    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









    $endgroup$



















      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      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











      $endgroup$












      • $begingroup$
        @ 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















      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











      $endgroup$












      • $begingroup$
        @ 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













      6












      6








      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











      $endgroup$



      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.







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited yesterday

























      answered yesterday









      José Carlos SantosJosé Carlos Santos

      168k22132236




      168k22132236











      • $begingroup$
        @ 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
















      • $begingroup$
        @ 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















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




      $begingroup$
      @ 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




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











      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









      $endgroup$

















        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









        $endgroup$















          6












          6








          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









          $endgroup$



          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.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered yesterday









          ArthurArthur

          118k7118201




          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









              $endgroup$

















                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









                $endgroup$















                  4












                  4








                  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









                  $endgroup$



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







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered yesterday









                  FredFred

                  48.3k1849




                  48.3k1849













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