$(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$ iff $X$ is countable.
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|$$=n$, then the subspace $A={1,2,3,...,n}$ with the induced Euclidean topology is homeomorphic to $(X, tau)$. If X is countably infinite, then the subspace $mathbb{N} $ is homeomophic to $(X, tau)$.
Now how we can prove the other direction, i.e., if $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$, then $X$ is countable?
real-analysis general-topology metric-spaces
edited Nov 21 '18 at 18:07
Cassius12
11611
asked Jan 6 '17 at 18:05
amir bahadory
1,253417
|
show 1 more comment
Let $mathbb{R}$ have the Euclidean topology (i.e., with Euclidean distance), and let $(X, tau)$ be a discrete topological space.
Prove that $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$ iff $X$ is countable.
Let $X$ be countable. If $|X|$$=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
edited Nov 21 '18 at 18:07
Cassius12
11611
asked Jan 6 '17 at 18:05
amir bahadory
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
|
show 1 more comment
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|$$=n$, then the subspace $A={1,2,3,...,n}$ with the induced Euclidean topology is homeomorphic to $(X, tau)$. If X is countably infinite, then the subspace $mathbb{N} $ is homeomophic to $(X, tau)$.
Now how we can prove the other direction, i.e., if $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$, then $X$ is countable?
real-analysis general-topology metric-spaces
edited Nov 21 '18 at 18:07
Cassius12
11611
asked Jan 6 '17 at 18:05
amir bahadory
1,253417
Let $mathbb{R}$ have the Euclidean topology (i.e., with Euclidean distance), and let $(X, tau)$ be a discrete topological space.
Prove that $(X, tau)$ is homeomorphic to a subspace of $mathbb{R}$ iff $X$ is countable.
Let $X$ be countable. If $|X|$$=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
edited Nov 21 '18 at 18:07
Cassius12
11611
asked Jan 6 '17 at 18:05
amir bahadory
1,253417
edited Nov 21 '18 at 18:07
Cassius12
11611
asked Jan 6 '17 at 18:05
amir bahadory
1,253417
edited Nov 21 '18 at 18:07
Cassius12
11611
edited Nov 21 '18 at 18:07
Cassius12
11611
edited Nov 21 '18 at 18:07
Cassius12
11611
11611
asked Jan 6 '17 at 18:05
amir bahadory
1,253417
asked Jan 6 '17 at 18:05
amir bahadory
1,253417
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
|
show 1 more comment
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
|
show 1 more comment
1 Answer
1
active
oldest
votes
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.
answered Jan 6 '17 at 21:00
Brian M. Scott
455k38505907
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2086492%2fx-tau-is-homeomorphic-to-a-subspace-of-mathbbr-iff-x-is-countable%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
answered Jan 6 '17 at 21:00
Brian M. Scott
455k38505907
add a comment |
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.
answered Jan 6 '17 at 21:00
Brian M. Scott
455k38505907
add a comment |
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.
answered Jan 6 '17 at 21:00
Brian M. Scott
455k38505907
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.
answered Jan 6 '17 at 21:00
Brian M. Scott
455k38505907
answered Jan 6 '17 at 21:00
Brian M. Scott
455k38505907
answered Jan 6 '17 at 21:00
Brian M. Scott
455k38505907
answered Jan 6 '17 at 21:00
Brian M. Scott
455k38505907
455k38505907
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2086492%2fx-tau-is-homeomorphic-to-a-subspace-of-mathbbr-iff-x-is-countable%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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