Uniform continuity and compactness












4














We know that, if a function $f$ is continuous mapping from a compact metric space to another metric space say $Y$ then $f$ is uniformly continuous.



Do we have a generalization of this theorem for general topological space.






general-topology continuity compactness uniform-continuity







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  • 1




    I think there is none because we cannot define uniform continuity properly without the metric.
    – Song
    2 days ago










  • It's not true for non-compact spaces + see @Song 's comment
    – Yanko
    2 days ago








  • 6




    To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
    – mouthetics
    2 days ago








  • 1




    @StammeringMathematician You're welcome. Happy new year.
    – mouthetics
    2 days ago






  • 1




    @mouthetics I also learned there is uniform structure from you comment. Thank you!
    – Song
    2 days ago
















4














We know that, if a function $f$ is continuous mapping from a compact metric space to another metric space say $Y$ then $f$ is uniformly continuous.



Do we have a generalization of this theorem for general topological space.






general-topology continuity compactness uniform-continuity







share|cite|improve this question




















  • 1




    I think there is none because we cannot define uniform continuity properly without the metric.
    – Song
    2 days ago










  • It's not true for non-compact spaces + see @Song 's comment
    – Yanko
    2 days ago








  • 6




    To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
    – mouthetics
    2 days ago








  • 1




    @StammeringMathematician You're welcome. Happy new year.
    – mouthetics
    2 days ago






  • 1




    @mouthetics I also learned there is uniform structure from you comment. Thank you!
    – Song
    2 days ago














4












4








4







We know that, if a function $f$ is continuous mapping from a compact metric space to another metric space say $Y$ then $f$ is uniformly continuous.



Do we have a generalization of this theorem for general topological space.






general-topology continuity compactness uniform-continuity







share|cite|improve this question















We know that, if a function $f$ is continuous mapping from a compact metric space to another metric space say $Y$ then $f$ is uniformly continuous.



Do we have a generalization of this theorem for general topological space.






general-topology continuity compactness uniform-continuity




general-topology continuity compactness uniform-continuity






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Scientifica

6,37641335




6,37641335










asked 2 days ago









StammeringMathematician

2,2381322




2,2381322








  • 1




    I think there is none because we cannot define uniform continuity properly without the metric.
    – Song
    2 days ago










  • It's not true for non-compact spaces + see @Song 's comment
    – Yanko
    2 days ago








  • 6




    To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
    – mouthetics
    2 days ago








  • 1




    @StammeringMathematician You're welcome. Happy new year.
    – mouthetics
    2 days ago






  • 1




    @mouthetics I also learned there is uniform structure from you comment. Thank you!
    – Song
    2 days ago














  • 1




    I think there is none because we cannot define uniform continuity properly without the metric.
    – Song
    2 days ago










  • It's not true for non-compact spaces + see @Song 's comment
    – Yanko
    2 days ago








  • 6




    To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
    – mouthetics
    2 days ago








  • 1




    @StammeringMathematician You're welcome. Happy new year.
    – mouthetics
    2 days ago






  • 1




    @mouthetics I also learned there is uniform structure from you comment. Thank you!
    – Song
    2 days ago








1




1




I think there is none because we cannot define uniform continuity properly without the metric.
– Song
2 days ago




I think there is none because we cannot define uniform continuity properly without the metric.
– Song
2 days ago












It's not true for non-compact spaces + see @Song 's comment
– Yanko
2 days ago






It's not true for non-compact spaces + see @Song 's comment
– Yanko
2 days ago






6




6




To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
– mouthetics
2 days ago






To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
– mouthetics
2 days ago






1




1




@StammeringMathematician You're welcome. Happy new year.
– mouthetics
2 days ago




@StammeringMathematician You're welcome. Happy new year.
– mouthetics
2 days ago




1




1




@mouthetics I also learned there is uniform structure from you comment. Thank you!
– Song
2 days ago




@mouthetics I also learned there is uniform structure from you comment. Thank you!
– Song
2 days ago










1 Answer
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4














To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.



Here's the definition of uniform continuity for metric spaces:




Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.



A function $f:Xto Y$ is uniformly continuous if



$$forallvarepsilon>0,existsdelta>0,forall x,yin X, d_X(x,y)<deltaimplies d_Y(f(x),f(y))<varepsilon.$$




Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.



Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.



Your result generalizes to this:




Let $X$, $Y$ be uniform spaces and $f:Xto Y$ be a continuous function.



Suppose that $X$ is compact. Then $f$ is uniformly continuous.




I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.






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    4














    To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.



    Here's the definition of uniform continuity for metric spaces:




    Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.



    A function $f:Xto Y$ is uniformly continuous if



    $$forallvarepsilon>0,existsdelta>0,forall x,yin X, d_X(x,y)<deltaimplies d_Y(f(x),f(y))<varepsilon.$$




    Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.



    Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.



    Your result generalizes to this:




    Let $X$, $Y$ be uniform spaces and $f:Xto Y$ be a continuous function.



    Suppose that $X$ is compact. Then $f$ is uniformly continuous.




    I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.






    share|cite|improve this answer




























      4














      To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.



      Here's the definition of uniform continuity for metric spaces:




      Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.



      A function $f:Xto Y$ is uniformly continuous if



      $$forallvarepsilon>0,existsdelta>0,forall x,yin X, d_X(x,y)<deltaimplies d_Y(f(x),f(y))<varepsilon.$$




      Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.



      Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.



      Your result generalizes to this:




      Let $X$, $Y$ be uniform spaces and $f:Xto Y$ be a continuous function.



      Suppose that $X$ is compact. Then $f$ is uniformly continuous.




      I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.






      share|cite|improve this answer


























        4












        4








        4






        To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.



        Here's the definition of uniform continuity for metric spaces:




        Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.



        A function $f:Xto Y$ is uniformly continuous if



        $$forallvarepsilon>0,existsdelta>0,forall x,yin X, d_X(x,y)<deltaimplies d_Y(f(x),f(y))<varepsilon.$$




        Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.



        Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.



        Your result generalizes to this:




        Let $X$, $Y$ be uniform spaces and $f:Xto Y$ be a continuous function.



        Suppose that $X$ is compact. Then $f$ is uniformly continuous.




        I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.






        share|cite|improve this answer














        To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.



        Here's the definition of uniform continuity for metric spaces:




        Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.



        A function $f:Xto Y$ is uniformly continuous if



        $$forallvarepsilon>0,existsdelta>0,forall x,yin X, d_X(x,y)<deltaimplies d_Y(f(x),f(y))<varepsilon.$$




        Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.



        Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.



        Your result generalizes to this:




        Let $X$, $Y$ be uniform spaces and $f:Xto Y$ be a continuous function.



        Suppose that $X$ is compact. Then $f$ is uniformly continuous.




        I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








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        Henno Brandsma

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        Scientifica

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