Existence of an homeomorphic between [0,1] to X × YIs $[0,1]^omega$ 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
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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;
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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
$endgroup$
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
$endgroup$
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
$endgroup$
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
general-topology
asked Jun 11 at 8:25
WorriorWorrior
354
354
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add a comment |
2 Answers
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$begingroup$
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.)
$endgroup$
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.
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
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.)
$endgroup$
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.)
$endgroup$
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.)
$endgroup$
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.)
answered Jun 11 at 8:39
PedroPedro
3,0791722
3,0791722
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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.
$endgroup$
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.
$endgroup$
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.
$endgroup$
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.
answered Jun 11 at 8:32
José Carlos SantosJosé Carlos Santos
193k24148266
193k24148266
add a comment |
add a comment |
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