$(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$ iff $X$ is countable.












0














Let $mathbb{R}$ have the Euclidean topology (i.e., with Euclidean distance), and let $(X, tau)$ be a discrete topological space.



Prove that $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$ iff $X$ is countable.



Let $X$ be countable. If $|X|$$=n$, then the subspace $A={1,2,3,...,n}$ with the induced Euclidean topology is homeomorphic to $(X, tau)$. If X is countably infinite, then the subspace $mathbb{N} $ is homeomophic to $(X, tau)$.



Now how we can prove the other direction, i.e., if $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$, then $X$ is countable?






real-analysis general-topology metric-spaces







share|cite|improve this question
























  • As it's written right now the question is not understandable (for me), could you clarify what you're asking?
    – Alessandro Codenotti
    Jan 6 '17 at 18:20










  • if $B subset mathbb{R} $ with Euclidean distance be homeomorphic with (X,T) that T is discrete topology then prove X is countable .
    – amir bahadory
    Jan 6 '17 at 18:23












  • Are you asking "Prove that every discrete subspace of $mathbb{R}$ is countable"?
    – Dan Rust
    Jan 6 '17 at 18:25












  • I think he means: prove that there is a subspace $A$ of $mathbb R$ such that a subspace of $A$ is discrete if and only if it is countable.
    – Jorge Fernández
    Jan 6 '17 at 18:28












  • @AlessandroCodenotti. yes.
    – amir bahadory
    Jan 6 '17 at 18:28
















0














Let $mathbb{R}$ have the Euclidean topology (i.e., with Euclidean distance), and let $(X, tau)$ be a discrete topological space.



Prove that $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$ iff $X$ is countable.



Let $X$ be countable. If $|X|$$=n$, then the subspace $A={1,2,3,...,n}$ with the induced Euclidean topology is homeomorphic to $(X, tau)$. If X is countably infinite, then the subspace $mathbb{N} $ is homeomophic to $(X, tau)$.



Now how we can prove the other direction, i.e., if $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$, then $X$ is countable?






real-analysis general-topology metric-spaces







share|cite|improve this question
























  • As it's written right now the question is not understandable (for me), could you clarify what you're asking?
    – Alessandro Codenotti
    Jan 6 '17 at 18:20










  • if $B subset mathbb{R} $ with Euclidean distance be homeomorphic with (X,T) that T is discrete topology then prove X is countable .
    – amir bahadory
    Jan 6 '17 at 18:23












  • Are you asking "Prove that every discrete subspace of $mathbb{R}$ is countable"?
    – Dan Rust
    Jan 6 '17 at 18:25












  • I think he means: prove that there is a subspace $A$ of $mathbb R$ such that a subspace of $A$ is discrete if and only if it is countable.
    – Jorge Fernández
    Jan 6 '17 at 18:28












  • @AlessandroCodenotti. yes.
    – amir bahadory
    Jan 6 '17 at 18:28














0












0








0







Let $mathbb{R}$ have the Euclidean topology (i.e., with Euclidean distance), and let $(X, tau)$ be a discrete topological space.



Prove that $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$ iff $X$ is countable.



Let $X$ be countable. If $|X|$$=n$, then the subspace $A={1,2,3,...,n}$ with the induced Euclidean topology is homeomorphic to $(X, tau)$. If X is countably infinite, then the subspace $mathbb{N} $ is homeomophic to $(X, tau)$.



Now how we can prove the other direction, i.e., if $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$, then $X$ is countable?






real-analysis general-topology metric-spaces







share|cite|improve this question















Let $mathbb{R}$ have the Euclidean topology (i.e., with Euclidean distance), and let $(X, tau)$ be a discrete topological space.



Prove that $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$ iff $X$ is countable.



Let $X$ be countable. If $|X|$$=n$, then the subspace $A={1,2,3,...,n}$ with the induced Euclidean topology is homeomorphic to $(X, tau)$. If X is countably infinite, then the subspace $mathbb{N} $ is homeomophic to $(X, tau)$.



Now how we can prove the other direction, i.e., if $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$, then $X$ is countable?






real-analysis general-topology metric-spaces




real-analysis general-topology metric-spaces






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 21 '18 at 18:07









Cassius12

11611




11611










asked Jan 6 '17 at 18:05









amir bahadory

1,253417




1,253417












  • As it's written right now the question is not understandable (for me), could you clarify what you're asking?
    – Alessandro Codenotti
    Jan 6 '17 at 18:20










  • if $B subset mathbb{R} $ with Euclidean distance be homeomorphic with (X,T) that T is discrete topology then prove X is countable .
    – amir bahadory
    Jan 6 '17 at 18:23












  • Are you asking "Prove that every discrete subspace of $mathbb{R}$ is countable"?
    – Dan Rust
    Jan 6 '17 at 18:25












  • I think he means: prove that there is a subspace $A$ of $mathbb R$ such that a subspace of $A$ is discrete if and only if it is countable.
    – Jorge Fernández
    Jan 6 '17 at 18:28












  • @AlessandroCodenotti. yes.
    – amir bahadory
    Jan 6 '17 at 18:28


















  • As it's written right now the question is not understandable (for me), could you clarify what you're asking?
    – Alessandro Codenotti
    Jan 6 '17 at 18:20










  • if $B subset mathbb{R} $ with Euclidean distance be homeomorphic with (X,T) that T is discrete topology then prove X is countable .
    – amir bahadory
    Jan 6 '17 at 18:23












  • Are you asking "Prove that every discrete subspace of $mathbb{R}$ is countable"?
    – Dan Rust
    Jan 6 '17 at 18:25












  • I think he means: prove that there is a subspace $A$ of $mathbb R$ such that a subspace of $A$ is discrete if and only if it is countable.
    – Jorge Fernández
    Jan 6 '17 at 18:28












  • @AlessandroCodenotti. yes.
    – amir bahadory
    Jan 6 '17 at 18:28
















As it's written right now the question is not understandable (for me), could you clarify what you're asking?
– Alessandro Codenotti
Jan 6 '17 at 18:20




As it's written right now the question is not understandable (for me), could you clarify what you're asking?
– Alessandro Codenotti
Jan 6 '17 at 18:20












if $B subset mathbb{R} $ with Euclidean distance be homeomorphic with (X,T) that T is discrete topology then prove X is countable .
– amir bahadory
Jan 6 '17 at 18:23






if $B subset mathbb{R} $ with Euclidean distance be homeomorphic with (X,T) that T is discrete topology then prove X is countable .
– amir bahadory
Jan 6 '17 at 18:23














Are you asking "Prove that every discrete subspace of $mathbb{R}$ is countable"?
– Dan Rust
Jan 6 '17 at 18:25






Are you asking "Prove that every discrete subspace of $mathbb{R}$ is countable"?
– Dan Rust
Jan 6 '17 at 18:25














I think he means: prove that there is a subspace $A$ of $mathbb R$ such that a subspace of $A$ is discrete if and only if it is countable.
– Jorge Fernández
Jan 6 '17 at 18:28






I think he means: prove that there is a subspace $A$ of $mathbb R$ such that a subspace of $A$ is discrete if and only if it is countable.
– Jorge Fernández
Jan 6 '17 at 18:28














@AlessandroCodenotti. yes.
– amir bahadory
Jan 6 '17 at 18:28




@AlessandroCodenotti. yes.
– amir bahadory
Jan 6 '17 at 18:28










1 Answer
1






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0














HINT: Prove the contrapositive: show that if $AsubseteqBbb R$ is uncountable, then the relative topology on $A$ is not discrete. One way to do this is to let $mathscr{B}$ be a countable base for the topology of $Bbb R$, and let



$$mathscr{B}_0={Binmathscr{B}:Bcap Atext{ is countable}};.$$




  • Show that $Acapbigcupmathscr{B}_0$ is countable.


Let $A_0=Asetminusbigcupmathscr{B}_0$.




  • Show that $Anevarnothing$, and that each $xin A_0$ is a limit point of $A$. Conclude that $A$ is not discrete in its relative topology.






share|cite|improve this answer





















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    HINT: Prove the contrapositive: show that if $AsubseteqBbb R$ is uncountable, then the relative topology on $A$ is not discrete. One way to do this is to let $mathscr{B}$ be a countable base for the topology of $Bbb R$, and let



    $$mathscr{B}_0={Binmathscr{B}:Bcap Atext{ is countable}};.$$




    • Show that $Acapbigcupmathscr{B}_0$ is countable.


    Let $A_0=Asetminusbigcupmathscr{B}_0$.




    • Show that $Anevarnothing$, and that each $xin A_0$ is a limit point of $A$. Conclude that $A$ is not discrete in its relative topology.






    share|cite|improve this answer


























      0














      HINT: Prove the contrapositive: show that if $AsubseteqBbb R$ is uncountable, then the relative topology on $A$ is not discrete. One way to do this is to let $mathscr{B}$ be a countable base for the topology of $Bbb R$, and let



      $$mathscr{B}_0={Binmathscr{B}:Bcap Atext{ is countable}};.$$




      • Show that $Acapbigcupmathscr{B}_0$ is countable.


      Let $A_0=Asetminusbigcupmathscr{B}_0$.




      • Show that $Anevarnothing$, and that each $xin A_0$ is a limit point of $A$. Conclude that $A$ is not discrete in its relative topology.






      share|cite|improve this answer
























        0












        0








        0






        HINT: Prove the contrapositive: show that if $AsubseteqBbb R$ is uncountable, then the relative topology on $A$ is not discrete. One way to do this is to let $mathscr{B}$ be a countable base for the topology of $Bbb R$, and let



        $$mathscr{B}_0={Binmathscr{B}:Bcap Atext{ is countable}};.$$




        • Show that $Acapbigcupmathscr{B}_0$ is countable.


        Let $A_0=Asetminusbigcupmathscr{B}_0$.




        • Show that $Anevarnothing$, and that each $xin A_0$ is a limit point of $A$. Conclude that $A$ is not discrete in its relative topology.






        share|cite|improve this answer












        HINT: Prove the contrapositive: show that if $AsubseteqBbb R$ is uncountable, then the relative topology on $A$ is not discrete. One way to do this is to let $mathscr{B}$ be a countable base for the topology of $Bbb R$, and let



        $$mathscr{B}_0={Binmathscr{B}:Bcap Atext{ is countable}};.$$




        • Show that $Acapbigcupmathscr{B}_0$ is countable.


        Let $A_0=Asetminusbigcupmathscr{B}_0$.




        • Show that $Anevarnothing$, and that each $xin A_0$ is a limit point of $A$. Conclude that $A$ is not discrete in its relative topology.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 6 '17 at 21:00









        Brian M. Scott

        455k38505907




        455k38505907






























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