Open neighborhoods in a topological subspace
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Suppose we have a topological space $X$, an element $xin X$, and an open neighborhood $x in O subset X$. Further, suppose that $X$ is a topological subspace of $Y$.
I am trying to figure out under what conditions there must exist an open neighborhood of $x$ in $Y$?
general-topology
asked Nov 23 '18 at 18:36
DarrenDarren
345110
$endgroup$
add a comment |
$begingroup$
Suppose we have a topological space $X$, an element $xin X$, and an open neighborhood $x in O subset X$. Further, suppose that $X$ is a topological subspace of $Y$.
I am trying to figure out under what conditions there must exist an open neighborhood of $x$ in $Y$?
general-topology
asked Nov 23 '18 at 18:36
DarrenDarren
345110
$endgroup$
1
$begingroup$
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
$endgroup$
– WhatToDo
Nov 23 '18 at 19:00
1
$begingroup$
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
$endgroup$
– Federico
Nov 23 '18 at 19:05
$begingroup$
Induced topology
$endgroup$
– Avinash N
Nov 23 '18 at 19:40
add a comment |
$begingroup$
Suppose we have a topological space $X$, an element $xin X$, and an open neighborhood $x in O subset X$. Further, suppose that $X$ is a topological subspace of $Y$.
I am trying to figure out under what conditions there must exist an open neighborhood of $x$ in $Y$?
general-topology
asked Nov 23 '18 at 18:36
DarrenDarren
345110
$endgroup$
Suppose we have a topological space $X$, an element $xin X$, and an open neighborhood $x in O subset X$. Further, suppose that $X$ is a topological subspace of $Y$.
I am trying to figure out under what conditions there must exist an open neighborhood of $x$ in $Y$?
general-topology
general-topology
asked Nov 23 '18 at 18:36
DarrenDarren
345110
asked Nov 23 '18 at 18:36
DarrenDarren
345110
asked Nov 23 '18 at 18:36
DarrenDarren
345110
asked Nov 23 '18 at 18:36
DarrenDarren
345110
asked Nov 23 '18 at 18:36
DarrenDarren
345110
345110
1
$begingroup$
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
$endgroup$
– WhatToDo
Nov 23 '18 at 19:00
1
$begingroup$
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
$endgroup$
– Federico
Nov 23 '18 at 19:05
$begingroup$
Induced topology
$endgroup$
– Avinash N
Nov 23 '18 at 19:40
add a comment |
1
$begingroup$
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
$endgroup$
– WhatToDo
Nov 23 '18 at 19:00
1
$begingroup$
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
$endgroup$
– Federico
Nov 23 '18 at 19:05
$begingroup$
Induced topology
$endgroup$
– Avinash N
Nov 23 '18 at 19:40
1
1
$begingroup$
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
$endgroup$
– WhatToDo
Nov 23 '18 at 19:00
$begingroup$
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
$endgroup$
– WhatToDo
Nov 23 '18 at 19:00
1
1
$begingroup$
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
$endgroup$
– Federico
Nov 23 '18 at 19:05
$begingroup$
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
$endgroup$
– Federico
Nov 23 '18 at 19:05
$begingroup$
Induced topology
$endgroup$
– Avinash N
Nov 23 '18 at 19:40
$begingroup$
Induced topology
$endgroup$
– Avinash N
Nov 23 '18 at 19:40
add a comment |
1 Answer
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$begingroup$
ok, so the question as written is trivial since $Y$ is open.
The requirement $O = U cap X$ for some $U subset Y$ open is also true by definition of relative topology.
Thanks.
answered Nov 23 '18 at 19:41
DarrenDarren
345110
$endgroup$
add a comment |
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$begingroup$
ok, so the question as written is trivial since $Y$ is open.
The requirement $O = U cap X$ for some $U subset Y$ open is also true by definition of relative topology.
Thanks.
answered Nov 23 '18 at 19:41
DarrenDarren
345110
$endgroup$
add a comment |
$begingroup$
ok, so the question as written is trivial since $Y$ is open.
The requirement $O = U cap X$ for some $U subset Y$ open is also true by definition of relative topology.
Thanks.
answered Nov 23 '18 at 19:41
DarrenDarren
345110
$endgroup$
add a comment |
$begingroup$
ok, so the question as written is trivial since $Y$ is open.
The requirement $O = U cap X$ for some $U subset Y$ open is also true by definition of relative topology.
Thanks.
answered Nov 23 '18 at 19:41
DarrenDarren
345110
$endgroup$
ok, so the question as written is trivial since $Y$ is open.
The requirement $O = U cap X$ for some $U subset Y$ open is also true by definition of relative topology.
Thanks.
answered Nov 23 '18 at 19:41
DarrenDarren
345110
answered Nov 23 '18 at 19:41
DarrenDarren
345110
answered Nov 23 '18 at 19:41
DarrenDarren
345110
answered Nov 23 '18 at 19:41
DarrenDarren
345110
345110
add a comment |
add a comment |
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1
$begingroup$
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
$endgroup$
– WhatToDo
Nov 23 '18 at 19:00
1
$begingroup$
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
$endgroup$
– Federico
Nov 23 '18 at 19:05
$begingroup$
Induced topology
$endgroup$
– Avinash N
Nov 23 '18 at 19:40