Mixing
Local homeomorphism from a locally Euclidean space implies Euclidean space
2
$begingroup$
I'm trying to show that for $f:X rightarrow Y$ local homeomorphism and surjective If $X$ is locally Euclidean so is $Y$.
From Local homeomorphism we have that there exists $V_{x}$ such that $f:V_{x} rightarrow f(V_{x})$ where $y=f(x)$
But what If the neighbourhood $U_{x}$ from locally Euclidean is not contained in $V_{x}$?
general-topology
asked Jan 23 at 18:53
Daniel MoraesDaniel Moraes
326110
$endgroup$
add a comment |
2
$begingroup$
I'm trying to show that for $f:X rightarrow Y$ local homeomorphism and surjective If $X$ is locally Euclidean so is $Y$.
From Local homeomorphism we have that there exists $V_{x}$ such that $f:V_{x} rightarrow f(V_{x})$ where $y=f(x)$
But what If the neighbourhood $U_{x}$ from locally Euclidean is not contained in $V_{x}$?
general-topology
asked Jan 23 at 18:53
Daniel MoraesDaniel Moraes
326110
$endgroup$
add a comment |
2
2
2
$begingroup$
I'm trying to show that for $f:X rightarrow Y$ local homeomorphism and surjective If $X$ is locally Euclidean so is $Y$.
From Local homeomorphism we have that there exists $V_{x}$ such that $f:V_{x} rightarrow f(V_{x})$ where $y=f(x)$
But what If the neighbourhood $U_{x}$ from locally Euclidean is not contained in $V_{x}$?
general-topology
asked Jan 23 at 18:53
Daniel MoraesDaniel Moraes
326110
$endgroup$
I'm trying to show that for $f:X rightarrow Y$ local homeomorphism and surjective If $X$ is locally Euclidean so is $Y$.
From Local homeomorphism we have that there exists $V_{x}$ such that $f:V_{x} rightarrow f(V_{x})$ where $y=f(x)$
But what If the neighbourhood $U_{x}$ from locally Euclidean is not contained in $V_{x}$?
general-topology
general-topology
asked Jan 23 at 18:53
Daniel MoraesDaniel Moraes
326110
asked Jan 23 at 18:53
Daniel MoraesDaniel Moraes
326110
asked Jan 23 at 18:53
Daniel MoraesDaniel Moraes
326110
asked Jan 23 at 18:53
Daniel MoraesDaniel Moraes
326110
asked Jan 23 at 18:53
Daniel MoraesDaniel Moraes
326110
326110
add a comment |
add a comment |
1 Answer
1
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oldest
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2
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Great! Then take $U_xcap V_x$.
answered Jan 23 at 19:21
ScientificaScientifica
6,82941335
$endgroup$
$begingroup$
Is It still homeomorphic to a ball?
$endgroup$
– Daniel Moraes
Jan 23 at 19:29
1
$begingroup$
What is your definition of a locally Euclidean space? One of the equivalent definitions is that it's a space $X$ such that for every $xin X$ there exists an open set $U$ containing $x$ such that $U$ is homeomorphic to a Euclidean space (or to an open subset of a Euclidean space). $U_xcap V_x$ is an open subset of $U_x$ which is homeomorphic to an open subset of a Euclidean space (by definition of $U_x$ I guess). So $U_xcap V_x$ is homeomorphic to an open subset $W$ of a Euclidean space.
$endgroup$
– Scientifica
Jan 23 at 19:39
$begingroup$
If your definition requires an open ball, it's ok. Take an open ball in $W$.
$endgroup$
– Scientifica
Jan 23 at 19:42
$begingroup$
Not requiring an open ball, It is equivalent
$endgroup$
– Daniel Moraes
Jan 23 at 19:44
$begingroup$
I was stuck on seing It being open but i guees It is quite natural so
$endgroup$
– Daniel Moraes
Jan 23 at 19:45
|
show 2 more comments
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
2
$begingroup$
Great! Then take $U_xcap V_x$.
answered Jan 23 at 19:21
ScientificaScientifica
6,82941335
$endgroup$
$begingroup$
Is It still homeomorphic to a ball?
$endgroup$
– Daniel Moraes
Jan 23 at 19:29
1
$begingroup$
What is your definition of a locally Euclidean space? One of the equivalent definitions is that it's a space $X$ such that for every $xin X$ there exists an open set $U$ containing $x$ such that $U$ is homeomorphic to a Euclidean space (or to an open subset of a Euclidean space). $U_xcap V_x$ is an open subset of $U_x$ which is homeomorphic to an open subset of a Euclidean space (by definition of $U_x$ I guess). So $U_xcap V_x$ is homeomorphic to an open subset $W$ of a Euclidean space.
$endgroup$
– Scientifica
Jan 23 at 19:39
$begingroup$
If your definition requires an open ball, it's ok. Take an open ball in $W$.
$endgroup$
– Scientifica
Jan 23 at 19:42
$begingroup$
Not requiring an open ball, It is equivalent
$endgroup$
– Daniel Moraes
Jan 23 at 19:44
$begingroup$
I was stuck on seing It being open but i guees It is quite natural so
$endgroup$
– Daniel Moraes
Jan 23 at 19:45
|
show 2 more comments
2
$begingroup$
Great! Then take $U_xcap V_x$.
answered Jan 23 at 19:21
ScientificaScientifica
6,82941335
$endgroup$
$begingroup$
Is It still homeomorphic to a ball?
$endgroup$
– Daniel Moraes
Jan 23 at 19:29
1
$begingroup$
What is your definition of a locally Euclidean space? One of the equivalent definitions is that it's a space $X$ such that for every $xin X$ there exists an open set $U$ containing $x$ such that $U$ is homeomorphic to a Euclidean space (or to an open subset of a Euclidean space). $U_xcap V_x$ is an open subset of $U_x$ which is homeomorphic to an open subset of a Euclidean space (by definition of $U_x$ I guess). So $U_xcap V_x$ is homeomorphic to an open subset $W$ of a Euclidean space.
$endgroup$
– Scientifica
Jan 23 at 19:39
$begingroup$
If your definition requires an open ball, it's ok. Take an open ball in $W$.
$endgroup$
– Scientifica
Jan 23 at 19:42
$begingroup$
Not requiring an open ball, It is equivalent
$endgroup$
– Daniel Moraes
Jan 23 at 19:44
$begingroup$
I was stuck on seing It being open but i guees It is quite natural so
$endgroup$
– Daniel Moraes
Jan 23 at 19:45
|
show 2 more comments
2
2
2
$begingroup$
Great! Then take $U_xcap V_x$.
answered Jan 23 at 19:21
ScientificaScientifica
6,82941335
$endgroup$
Great! Then take $U_xcap V_x$.
answered Jan 23 at 19:21
ScientificaScientifica
6,82941335
answered Jan 23 at 19:21
ScientificaScientifica
6,82941335
answered Jan 23 at 19:21
ScientificaScientifica
6,82941335
answered Jan 23 at 19:21
ScientificaScientifica
6,82941335
6,82941335
$begingroup$
Is It still homeomorphic to a ball?
$endgroup$
– Daniel Moraes
Jan 23 at 19:29
1
$begingroup$
What is your definition of a locally Euclidean space? One of the equivalent definitions is that it's a space $X$ such that for every $xin X$ there exists an open set $U$ containing $x$ such that $U$ is homeomorphic to a Euclidean space (or to an open subset of a Euclidean space). $U_xcap V_x$ is an open subset of $U_x$ which is homeomorphic to an open subset of a Euclidean space (by definition of $U_x$ I guess). So $U_xcap V_x$ is homeomorphic to an open subset $W$ of a Euclidean space.
$endgroup$
– Scientifica
Jan 23 at 19:39
$begingroup$
If your definition requires an open ball, it's ok. Take an open ball in $W$.
$endgroup$
– Scientifica
Jan 23 at 19:42
$begingroup$
Not requiring an open ball, It is equivalent
$endgroup$
– Daniel Moraes
Jan 23 at 19:44
$begingroup$
I was stuck on seing It being open but i guees It is quite natural so
$endgroup$
– Daniel Moraes
Jan 23 at 19:45
|
show 2 more comments
$begingroup$
Is It still homeomorphic to a ball?
$endgroup$
– Daniel Moraes
Jan 23 at 19:29
1
$begingroup$
What is your definition of a locally Euclidean space? One of the equivalent definitions is that it's a space $X$ such that for every $xin X$ there exists an open set $U$ containing $x$ such that $U$ is homeomorphic to a Euclidean space (or to an open subset of a Euclidean space). $U_xcap V_x$ is an open subset of $U_x$ which is homeomorphic to an open subset of a Euclidean space (by definition of $U_x$ I guess). So $U_xcap V_x$ is homeomorphic to an open subset $W$ of a Euclidean space.
$endgroup$
– Scientifica
Jan 23 at 19:39
$begingroup$
If your definition requires an open ball, it's ok. Take an open ball in $W$.
$endgroup$
– Scientifica
Jan 23 at 19:42
$begingroup$
Not requiring an open ball, It is equivalent
$endgroup$
– Daniel Moraes
Jan 23 at 19:44
$begingroup$
I was stuck on seing It being open but i guees It is quite natural so
$endgroup$
– Daniel Moraes
Jan 23 at 19:45
$begingroup$
Is It still homeomorphic to a ball?
$endgroup$
– Daniel Moraes
Jan 23 at 19:29
$begingroup$
Is It still homeomorphic to a ball?
$endgroup$
– Daniel Moraes
Jan 23 at 19:29
1
1
$begingroup$
What is your definition of a locally Euclidean space? One of the equivalent definitions is that it's a space $X$ such that for every $xin X$ there exists an open set $U$ containing $x$ such that $U$ is homeomorphic to a Euclidean space (or to an open subset of a Euclidean space). $U_xcap V_x$ is an open subset of $U_x$ which is homeomorphic to an open subset of a Euclidean space (by definition of $U_x$ I guess). So $U_xcap V_x$ is homeomorphic to an open subset $W$ of a Euclidean space.
$endgroup$
– Scientifica
Jan 23 at 19:39
$begingroup$
What is your definition of a locally Euclidean space? One of the equivalent definitions is that it's a space $X$ such that for every $xin X$ there exists an open set $U$ containing $x$ such that $U$ is homeomorphic to a Euclidean space (or to an open subset of a Euclidean space). $U_xcap V_x$ is an open subset of $U_x$ which is homeomorphic to an open subset of a Euclidean space (by definition of $U_x$ I guess). So $U_xcap V_x$ is homeomorphic to an open subset $W$ of a Euclidean space.
$endgroup$
– Scientifica
Jan 23 at 19:39
$begingroup$
If your definition requires an open ball, it's ok. Take an open ball in $W$.
$endgroup$
– Scientifica
Jan 23 at 19:42
$begingroup$
If your definition requires an open ball, it's ok. Take an open ball in $W$.
$endgroup$
– Scientifica
Jan 23 at 19:42
$begingroup$
Not requiring an open ball, It is equivalent
$endgroup$
– Daniel Moraes
Jan 23 at 19:44
$begingroup$
Not requiring an open ball, It is equivalent
$endgroup$
– Daniel Moraes
Jan 23 at 19:44
$begingroup$
I was stuck on seing It being open but i guees It is quite natural so
$endgroup$
– Daniel Moraes
Jan 23 at 19:45
$begingroup$
I was stuck on seing It being open but i guees It is quite natural so
$endgroup$
– Daniel Moraes
Jan 23 at 19:45
|
show 2 more comments
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