Ttdfjt

How many people are necessary to maintain modern civilisation?

Proving an Intuitive Result Rigorously

"Correct me if I'm wrong"

Is there a name for the trope when there is a moments dialogue when someone pauses just before they leave the room?

Are there examples of rowers who also fought?

Print one file per line using echo

What was the flower of Empress Taytu?

Why isn't my calculation that we should be able to see the sun well beyond the observable universe valid?

Is the specular reflection on a polished gold sphere white or gold in colour?

I found a password with hashcat, but it doesn't work

How did the Vostok ejection seat safely eject an astronaut from a sealed space capsule?

Find the common ancestor between two nodes of a tree

How do I remove this inheritance-related code smell?

Methodology: Writing unit tests for another developer

Why don't countries like Japan just print more money?

What triggered jesuits' ban on infinitesimals in 1632?

What was the first third-party commercial application for MS-DOS?

How could empty set be unique if it could be vacuously false

What are the current battlegrounds for people’s “rights” in the UK?

Dmesg full of I/O errors, smart ok, four disks affected

How to work with PETG? Settings, caveats, etc

Greeting with "Ho"

Is there a term for the belief that "if it's legal, it's moral"?

When Bnei Yisroel travelled in the midbar, what happened on Shabbos?

Existence of an homeomorphic between [0,1] to X × Y

Is $[0,1]^omega$ homeomorphic to $D^omega$?Prove rigorously that for two points $x, y in M$, the spaces $M backslash x$ and $M backslash y$ are homeomorphic.Does the Stone-Čech compactification respect subspaces?How to show that $[0,1)$ and $(0,1]$ are or aren't homeomorphic with induced $mathcalT_l$ topologies?Are X and Y homeomorphic?Two topological spaces which imbed in each other and are quotients of each other but not homeomorphic?An example of non-homeomorphic surfaces, $S_1, S_2$, such that $S_1 times [0,1]$ is homeomorphic to $S_2 times [0,1]$construct two non-homeomorphic topological spacesComplete+bounded homeomorphic to incomplete+unboundeddistinction between homeomorphic topological spaces

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;

3

$begingroup$

I'm doing a practice exam questions and am stuck at this question:

Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?

I'm not even sure how to start tackle it, any help and clues will be appreciated! My head is leading me to "cut-points" area, but I'm not sure abuot it./

Thanks in advance!

general-topology

share|cite|improve this question

asked Jun 11 at 8:25

WorriorWorrior

354

$endgroup$

add a comment | 

3

$begingroup$

I'm doing a practice exam questions and am stuck at this question:

Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?

I'm not even sure how to start tackle it, any help and clues will be appreciated! My head is leading me to "cut-points" area, but I'm not sure abuot it./

Thanks in advance!

general-topology

share|cite|improve this question

asked Jun 11 at 8:25

WorriorWorrior

354

$endgroup$

add a comment | 

$begingroup$

I'm doing a practice exam questions and am stuck at this question:

Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?

I'm not even sure how to start tackle it, any help and clues will be appreciated! My head is leading me to "cut-points" area, but I'm not sure abuot it./

Thanks in advance!

general-topology

share|cite|improve this question

asked Jun 11 at 8:25

WorriorWorrior

354

$endgroup$

share|cite|improve this question

asked Jun 11 at 8:25

WorriorWorrior

354

share|cite|improve this question

asked Jun 11 at 8:25

WorriorWorrior

354

asked Jun 11 at 8:25

WorriorWorrior

354

asked Jun 11 at 8:25

WorriorWorrior

354

2 Answers
2

active

oldest

votes

7

$begingroup$

You had the right idea. $X$ and $Y$ are the image under the projections of $Xtimes Y$, so they must be path connected.

Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $Xtimes Y$, assuming both have more than one point?

(Hint: make a picture and try to connect two arbitrary points with a path.)

share|cite|improve this answer

answered Jun 11 at 8:39

PedroPedro

3,0791722

$endgroup$

add a comment | 

4

$begingroup$

Hints: Prove that:

• If $Xtimes Y$ is homeomorphic to $[0,1]$, then both $X$ and $Y$ are connected.


• If $x_0in X$ and $y_0in Y$, then $(Xtimes Y)setminus(x_0,y_0)$ is still connected.

share|cite|improve this answer

answered Jun 11 at 8:32

José Carlos SantosJosé Carlos Santos

193k24148266

$endgroup$

add a comment | 

Your Answer

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
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3258372%2fexistence-of-an-homeomorphic-between-0-1-to-x-%25c3%2597-y%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

7

$begingroup$

You had the right idea. $X$ and $Y$ are the image under the projections of $Xtimes Y$, so they must be path connected.

Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $Xtimes Y$, assuming both have more than one point?

(Hint: make a picture and try to connect two arbitrary points with a path.)

share|cite|improve this answer

answered Jun 11 at 8:39

PedroPedro

3,0791722

$endgroup$

add a comment | 

7

$begingroup$

You had the right idea. $X$ and $Y$ are the image under the projections of $Xtimes Y$, so they must be path connected.

Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $Xtimes Y$, assuming both have more than one point?

(Hint: make a picture and try to connect two arbitrary points with a path.)

share|cite|improve this answer

answered Jun 11 at 8:39

PedroPedro

3,0791722

$endgroup$

add a comment | 

$begingroup$

You had the right idea. $X$ and $Y$ are the image under the projections of $Xtimes Y$, so they must be path connected.

Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $Xtimes Y$, assuming both have more than one point?

(Hint: make a picture and try to connect two arbitrary points with a path.)

share|cite|improve this answer

answered Jun 11 at 8:39

PedroPedro

3,0791722

$endgroup$

share|cite|improve this answer

answered Jun 11 at 8:39

PedroPedro

3,0791722

answered Jun 11 at 8:39

PedroPedro

3,0791722

answered Jun 11 at 8:39

PedroPedro

3,0791722

4

$begingroup$

Hints: Prove that:

• If $Xtimes Y$ is homeomorphic to $[0,1]$, then both $X$ and $Y$ are connected.


• If $x_0in X$ and $y_0in Y$, then $(Xtimes Y)setminus(x_0,y_0)$ is still connected.

share|cite|improve this answer

answered Jun 11 at 8:32

José Carlos SantosJosé Carlos Santos

193k24148266

$endgroup$

add a comment | 

4

$begingroup$

Hints: Prove that:

• If $Xtimes Y$ is homeomorphic to $[0,1]$, then both $X$ and $Y$ are connected.


• If $x_0in X$ and $y_0in Y$, then $(Xtimes Y)setminus(x_0,y_0)$ is still connected.

share|cite|improve this answer

answered Jun 11 at 8:32

José Carlos SantosJosé Carlos Santos

193k24148266

$endgroup$

add a comment | 

$begingroup$

Hints: Prove that:

• If $Xtimes Y$ is homeomorphic to $[0,1]$, then both $X$ and $Y$ are connected.


• If $x_0in X$ and $y_0in Y$, then $(Xtimes Y)setminus(x_0,y_0)$ is still connected.

share|cite|improve this answer

answered Jun 11 at 8:32

José Carlos SantosJosé Carlos Santos

193k24148266

$endgroup$

share|cite|improve this answer

answered Jun 11 at 8:32

José Carlos SantosJosé Carlos Santos

193k24148266

answered Jun 11 at 8:32

José Carlos SantosJosé Carlos Santos

193k24148266

answered Jun 11 at 8:32

José Carlos SantosJosé Carlos Santos

193k24148266