let $X$ be an arbitrary Hausdroff space, then is $Xsetminus{x_0}$ always dense? [closed]
$begingroup$
I know $mathbb{R}setminus{x_0}$ is dense subset of $mathbb{R}$. Is this result true for any arbitrary Hausdroff Topological space.
I mean, let $X$ be an arbitrary Hausdroff space, then is $Xsetminus{x_0}$ always dense?
general-topology
asked Jan 15 at 10:09
StammeringMathematicianStammeringMathematician
2,8321324
$endgroup$
closed as off-topic by Namaste, Shailesh, Eevee Trainer, lEm, Cesareo Jan 16 at 7:21
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Namaste, Shailesh, Eevee Trainer, lEm, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I know $mathbb{R}setminus{x_0}$ is dense subset of $mathbb{R}$. Is this result true for any arbitrary Hausdroff Topological space.
I mean, let $X$ be an arbitrary Hausdroff space, then is $Xsetminus{x_0}$ always dense?
general-topology
asked Jan 15 at 10:09
StammeringMathematicianStammeringMathematician
2,8321324
$endgroup$
closed as off-topic by Namaste, Shailesh, Eevee Trainer, lEm, Cesareo Jan 16 at 7:21
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Namaste, Shailesh, Eevee Trainer, lEm, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I know $mathbb{R}setminus{x_0}$ is dense subset of $mathbb{R}$. Is this result true for any arbitrary Hausdroff Topological space.
I mean, let $X$ be an arbitrary Hausdroff space, then is $Xsetminus{x_0}$ always dense?
general-topology
asked Jan 15 at 10:09
StammeringMathematicianStammeringMathematician
2,8321324
$endgroup$
I know $mathbb{R}setminus{x_0}$ is dense subset of $mathbb{R}$. Is this result true for any arbitrary Hausdroff Topological space.
I mean, let $X$ be an arbitrary Hausdroff space, then is $Xsetminus{x_0}$ always dense?
general-topology
general-topology
asked Jan 15 at 10:09
StammeringMathematicianStammeringMathematician
2,8321324
asked Jan 15 at 10:09
StammeringMathematicianStammeringMathematician
2,8321324
asked Jan 15 at 10:09
StammeringMathematicianStammeringMathematician
2,8321324
asked Jan 15 at 10:09
StammeringMathematicianStammeringMathematician
2,8321324
asked Jan 15 at 10:09
StammeringMathematicianStammeringMathematician
2,8321324
2,8321324
closed as off-topic by Namaste, Shailesh, Eevee Trainer, lEm, Cesareo Jan 16 at 7:21
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Namaste, Shailesh, Eevee Trainer, lEm, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Namaste, Shailesh, Eevee Trainer, lEm, Cesareo Jan 16 at 7:21
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Namaste, Shailesh, Eevee Trainer, lEm, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
No. The set $Xsetminus{x_0}$ is dense if and only if $x_0$ is not an isolated point. So, take $mathbb Z$ with the usual topology, for instance, and any $x_0inmathbb Z$.
answered Jan 15 at 10:13
José Carlos SantosJosé Carlos Santos
177k24138248
$endgroup$
$begingroup$
Perfect answer. This clear all the air of doubts. Thanks.
$endgroup$
– StammeringMathematician
Jan 15 at 10:17
add a comment |
$begingroup$
No.
Take $X={x_1, x_2}$ with the discrete metric.
answered Jan 15 at 10:12
BigbearZzzBigbearZzz
9,08921753
$endgroup$
add a comment |
$begingroup$
It is not true. For example, taking $X=[0,1]cup{2}$, we can see that $Xsetminus{2}$ is not dense.
$Xsetminus {x_0}$ is dense if and only if its closure is $X$. This means that it is dense if and only if it is not closed.
Remember, in a Hausdorff space, every singleton is closed, which means $Xsetminus{x_0}$ must be open. This, along with the statement above, means that $Xsetminus{x_0}$ is dense if and only if $x_0$ is not an isolated point.
answered Jan 15 at 10:13
5xum5xum
92.8k395162
$endgroup$
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No. The set $Xsetminus{x_0}$ is dense if and only if $x_0$ is not an isolated point. So, take $mathbb Z$ with the usual topology, for instance, and any $x_0inmathbb Z$.
answered Jan 15 at 10:13
José Carlos SantosJosé Carlos Santos
177k24138248
$endgroup$
$begingroup$
Perfect answer. This clear all the air of doubts. Thanks.
$endgroup$
– StammeringMathematician
Jan 15 at 10:17
add a comment |
$begingroup$
No. The set $Xsetminus{x_0}$ is dense if and only if $x_0$ is not an isolated point. So, take $mathbb Z$ with the usual topology, for instance, and any $x_0inmathbb Z$.
answered Jan 15 at 10:13
José Carlos SantosJosé Carlos Santos
177k24138248
$endgroup$
$begingroup$
Perfect answer. This clear all the air of doubts. Thanks.
$endgroup$
– StammeringMathematician
Jan 15 at 10:17
add a comment |
$begingroup$
No. The set $Xsetminus{x_0}$ is dense if and only if $x_0$ is not an isolated point. So, take $mathbb Z$ with the usual topology, for instance, and any $x_0inmathbb Z$.
answered Jan 15 at 10:13
José Carlos SantosJosé Carlos Santos
177k24138248
$endgroup$
No. The set $Xsetminus{x_0}$ is dense if and only if $x_0$ is not an isolated point. So, take $mathbb Z$ with the usual topology, for instance, and any $x_0inmathbb Z$.
answered Jan 15 at 10:13
José Carlos SantosJosé Carlos Santos
177k24138248
answered Jan 15 at 10:13
José Carlos SantosJosé Carlos Santos
177k24138248
answered Jan 15 at 10:13
José Carlos SantosJosé Carlos Santos
177k24138248
answered Jan 15 at 10:13
José Carlos SantosJosé Carlos Santos
177k24138248
177k24138248
$begingroup$
Perfect answer. This clear all the air of doubts. Thanks.
$endgroup$
– StammeringMathematician
Jan 15 at 10:17
add a comment |
$begingroup$
Perfect answer. This clear all the air of doubts. Thanks.
$endgroup$
– StammeringMathematician
Jan 15 at 10:17
$begingroup$
Perfect answer. This clear all the air of doubts. Thanks.
$endgroup$
– StammeringMathematician
Jan 15 at 10:17
$begingroup$
Perfect answer. This clear all the air of doubts. Thanks.
$endgroup$
– StammeringMathematician
Jan 15 at 10:17
add a comment |
$begingroup$
No.
Take $X={x_1, x_2}$ with the discrete metric.
answered Jan 15 at 10:12
BigbearZzzBigbearZzz
9,08921753
$endgroup$
add a comment |
$begingroup$
No.
Take $X={x_1, x_2}$ with the discrete metric.
answered Jan 15 at 10:12
BigbearZzzBigbearZzz
9,08921753
$endgroup$
add a comment |
$begingroup$
No.
Take $X={x_1, x_2}$ with the discrete metric.
answered Jan 15 at 10:12
BigbearZzzBigbearZzz
9,08921753
$endgroup$
No.
Take $X={x_1, x_2}$ with the discrete metric.
answered Jan 15 at 10:12
BigbearZzzBigbearZzz
9,08921753
answered Jan 15 at 10:12
BigbearZzzBigbearZzz
9,08921753
answered Jan 15 at 10:12
BigbearZzzBigbearZzz
9,08921753
answered Jan 15 at 10:12
BigbearZzzBigbearZzz
9,08921753
9,08921753
add a comment |
add a comment |
$begingroup$
It is not true. For example, taking $X=[0,1]cup{2}$, we can see that $Xsetminus{2}$ is not dense.
$Xsetminus {x_0}$ is dense if and only if its closure is $X$. This means that it is dense if and only if it is not closed.
Remember, in a Hausdorff space, every singleton is closed, which means $Xsetminus{x_0}$ must be open. This, along with the statement above, means that $Xsetminus{x_0}$ is dense if and only if $x_0$ is not an isolated point.
answered Jan 15 at 10:13
5xum5xum
92.8k395162
$endgroup$
add a comment |
$begingroup$
It is not true. For example, taking $X=[0,1]cup{2}$, we can see that $Xsetminus{2}$ is not dense.
$Xsetminus {x_0}$ is dense if and only if its closure is $X$. This means that it is dense if and only if it is not closed.
Remember, in a Hausdorff space, every singleton is closed, which means $Xsetminus{x_0}$ must be open. This, along with the statement above, means that $Xsetminus{x_0}$ is dense if and only if $x_0$ is not an isolated point.
answered Jan 15 at 10:13
5xum5xum
92.8k395162
$endgroup$
add a comment |
$begingroup$
It is not true. For example, taking $X=[0,1]cup{2}$, we can see that $Xsetminus{2}$ is not dense.
$Xsetminus {x_0}$ is dense if and only if its closure is $X$. This means that it is dense if and only if it is not closed.
Remember, in a Hausdorff space, every singleton is closed, which means $Xsetminus{x_0}$ must be open. This, along with the statement above, means that $Xsetminus{x_0}$ is dense if and only if $x_0$ is not an isolated point.
answered Jan 15 at 10:13
5xum5xum
92.8k395162
$endgroup$
It is not true. For example, taking $X=[0,1]cup{2}$, we can see that $Xsetminus{2}$ is not dense.
$Xsetminus {x_0}$ is dense if and only if its closure is $X$. This means that it is dense if and only if it is not closed.
Remember, in a Hausdorff space, every singleton is closed, which means $Xsetminus{x_0}$ must be open. This, along with the statement above, means that $Xsetminus{x_0}$ is dense if and only if $x_0$ is not an isolated point.
answered Jan 15 at 10:13
5xum5xum
92.8k395162
answered Jan 15 at 10:13
5xum5xum
92.8k395162
answered Jan 15 at 10:13
5xum5xum
92.8k395162
answered Jan 15 at 10:13
5xum5xum
92.8k395162
92.8k395162
add a comment |
add a comment |
