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 at 18:36
Darren
345110
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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 at 18:36
Darren
345110
1
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
– WhatToDo
Nov 23 at 19:00
1
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
– Federico
Nov 23 at 19:05
Induced topology
– Avinash N
Nov 23 at 19:40
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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 at 18:36
Darren
345110
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 at 18:36
Darren
345110
asked Nov 23 at 18:36
Darren
345110
asked Nov 23 at 18:36
Darren
345110
asked Nov 23 at 18:36
Darren
345110
asked Nov 23 at 18:36
Darren
345110
345110
1
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
– WhatToDo
Nov 23 at 19:00
1
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
– Federico
Nov 23 at 19:05
Induced topology
– Avinash N
Nov 23 at 19:40
add a comment |
1
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
– WhatToDo
Nov 23 at 19:00
1
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
– Federico
Nov 23 at 19:05
Induced topology
– Avinash N
Nov 23 at 19:40
1
1
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
– WhatToDo
Nov 23 at 19:00
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
– WhatToDo
Nov 23 at 19:00
1
1
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
– Federico
Nov 23 at 19:05
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
– Federico
Nov 23 at 19:05
Induced topology
– Avinash N
Nov 23 at 19:40
Induced topology
– Avinash N
Nov 23 at 19:40
add a comment |
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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 at 19:41
Darren
345110
add a comment |
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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 at 19:41
Darren
345110
add a comment |
up vote
0
down vote
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 at 19:41
Darren
345110
add a comment |
up vote
0
down vote
up vote
0
down vote
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 at 19:41
Darren
345110
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 at 19:41
Darren
345110
answered Nov 23 at 19:41
Darren
345110
answered Nov 23 at 19:41
Darren
345110
answered Nov 23 at 19:41
Darren
345110
345110
add a comment |
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1
$Y$ is an open neighborhood of $x$. Are you sure you're phrasing the question right?
– WhatToDo
Nov 23 at 19:00
1
Maybe you want $O=Ucap X$ for some $Usubset Y$ open? But that is just the definition of subset topology
– Federico
Nov 23 at 19:05
Induced topology
– Avinash N
Nov 23 at 19:40