when are two topological spaces homeomorphic?homeomorphic quotient spacesDemonstrate Topological Spaces are HomeomorphicHow to see that two sets are homeomorphic with respect to the subspace topologyIf two topological spaces have the same topological properties, are they homeomorphic?Showing that 2 topological spaces are homeomorphicHomeomorphic products of topological spacesShowing two topological spaces are not homeomorphicQuestion of whether two given spaces are homeomorphic.construct two non-homeomorphic topological spacesWhen are quotients of homeomorphic spaces homeomorphic?
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when are two topological spaces homeomorphic?
homeomorphic quotient spacesDemonstrate Topological Spaces are HomeomorphicHow to see that two sets are homeomorphic with respect to the subspace topologyIf two topological spaces have the same topological properties, are they homeomorphic?Showing that 2 topological spaces are homeomorphicHomeomorphic products of topological spacesShowing two topological spaces are not homeomorphicQuestion of whether two given spaces are homeomorphic.construct two non-homeomorphic topological spacesWhen are quotients of homeomorphic spaces homeomorphic?
$begingroup$
X and Y are topological spaces. If X is homeomorphic to a subspace of Y and Y is homeomorphic to
a subspace of X then X and Y are homeomorphic. Is this statement true ? justify
I know this is true but am unable to justify it. I tried to construct a homeomorphism but failed.
general-topology
asked Apr 30 at 18:33
Souvik DebSouvik Deb
224
$endgroup$
add a comment |
$begingroup$
X and Y are topological spaces. If X is homeomorphic to a subspace of Y and Y is homeomorphic to
a subspace of X then X and Y are homeomorphic. Is this statement true ? justify
I know this is true but am unable to justify it. I tried to construct a homeomorphism but failed.
general-topology
asked Apr 30 at 18:33
Souvik DebSouvik Deb
224
$endgroup$
6
$begingroup$
What you know is false.
$endgroup$
– Moishe Kohan
Apr 30 at 18:36
add a comment |
$begingroup$
X and Y are topological spaces. If X is homeomorphic to a subspace of Y and Y is homeomorphic to
a subspace of X then X and Y are homeomorphic. Is this statement true ? justify
I know this is true but am unable to justify it. I tried to construct a homeomorphism but failed.
general-topology
asked Apr 30 at 18:33
Souvik DebSouvik Deb
224
$endgroup$
X and Y are topological spaces. If X is homeomorphic to a subspace of Y and Y is homeomorphic to
a subspace of X then X and Y are homeomorphic. Is this statement true ? justify
I know this is true but am unable to justify it. I tried to construct a homeomorphism but failed.
general-topology
general-topology
asked Apr 30 at 18:33
Souvik DebSouvik Deb
224
asked Apr 30 at 18:33
Souvik DebSouvik Deb
224
asked Apr 30 at 18:33
Souvik DebSouvik Deb
224
asked Apr 30 at 18:33
Souvik DebSouvik Deb
224
asked Apr 30 at 18:33
Souvik DebSouvik Deb
224
224
6
$begingroup$
What you know is false.
$endgroup$
– Moishe Kohan
Apr 30 at 18:36
add a comment |
6
$begingroup$
What you know is false.
$endgroup$
– Moishe Kohan
Apr 30 at 18:36
6
6
$begingroup$
What you know is false.
$endgroup$
– Moishe Kohan
Apr 30 at 18:36
$begingroup$
What you know is false.
$endgroup$
– Moishe Kohan
Apr 30 at 18:36
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$mathbbR$ is homeomorphic to (0,1) which is a subspace of $[-1,2]$ and $[-1,2]$ is homeomorphic to $[-1,2]$ which is a subspace of $mathbbR$. But $[-1,2]$ is compact and $mathbbR$ is not so they are not homeomorphic.
edited Apr 30 at 18:38
mihaild
2,409216
answered Apr 30 at 18:37
Adam ChalumeauAdam Chalumeau
859116
$endgroup$
add a comment |
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1 Answer
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$begingroup$
$mathbbR$ is homeomorphic to (0,1) which is a subspace of $[-1,2]$ and $[-1,2]$ is homeomorphic to $[-1,2]$ which is a subspace of $mathbbR$. But $[-1,2]$ is compact and $mathbbR$ is not so they are not homeomorphic.
edited Apr 30 at 18:38
mihaild
2,409216
answered Apr 30 at 18:37
Adam ChalumeauAdam Chalumeau
859116
$endgroup$
add a comment |
$begingroup$
$mathbbR$ is homeomorphic to (0,1) which is a subspace of $[-1,2]$ and $[-1,2]$ is homeomorphic to $[-1,2]$ which is a subspace of $mathbbR$. But $[-1,2]$ is compact and $mathbbR$ is not so they are not homeomorphic.
edited Apr 30 at 18:38
mihaild
2,409216
answered Apr 30 at 18:37
Adam ChalumeauAdam Chalumeau
859116
$endgroup$
add a comment |
$begingroup$
$mathbbR$ is homeomorphic to (0,1) which is a subspace of $[-1,2]$ and $[-1,2]$ is homeomorphic to $[-1,2]$ which is a subspace of $mathbbR$. But $[-1,2]$ is compact and $mathbbR$ is not so they are not homeomorphic.
edited Apr 30 at 18:38
mihaild
2,409216
answered Apr 30 at 18:37
Adam ChalumeauAdam Chalumeau
859116
$endgroup$
$mathbbR$ is homeomorphic to (0,1) which is a subspace of $[-1,2]$ and $[-1,2]$ is homeomorphic to $[-1,2]$ which is a subspace of $mathbbR$. But $[-1,2]$ is compact and $mathbbR$ is not so they are not homeomorphic.
edited Apr 30 at 18:38
mihaild
2,409216
answered Apr 30 at 18:37
Adam ChalumeauAdam Chalumeau
859116
edited Apr 30 at 18:38
mihaild
2,409216
edited Apr 30 at 18:38
mihaild
2,409216
edited Apr 30 at 18:38
mihaild
2,409216
2,409216
answered Apr 30 at 18:37
Adam ChalumeauAdam Chalumeau
859116
answered Apr 30 at 18:37
Adam ChalumeauAdam Chalumeau
859116
answered Apr 30 at 18:37
Adam ChalumeauAdam Chalumeau
859116
859116
add a comment |
add a comment |
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What you know is false.
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– Moishe Kohan
Apr 30 at 18:36