Topological Properties closed sets
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I'm slightly confused with the idea of 'topological properties'
Is closed-ness of a subset of a metric space X a topological property?
I think it is because if a subset is closed under 1 metric then it should be closed under another?
general-topology metric-spaces
asked Nov 15 at 12:13
MathematicianP
3315
add a comment |
up vote
0
down vote
favorite
I'm slightly confused with the idea of 'topological properties'
Is closed-ness of a subset of a metric space X a topological property?
I think it is because if a subset is closed under 1 metric then it should be closed under another?
general-topology metric-spaces
asked Nov 15 at 12:13
MathematicianP
3315
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm slightly confused with the idea of 'topological properties'
Is closed-ness of a subset of a metric space X a topological property?
I think it is because if a subset is closed under 1 metric then it should be closed under another?
general-topology metric-spaces
asked Nov 15 at 12:13
MathematicianP
3315
I'm slightly confused with the idea of 'topological properties'
Is closed-ness of a subset of a metric space X a topological property?
I think it is because if a subset is closed under 1 metric then it should be closed under another?
general-topology metric-spaces
general-topology metric-spaces
asked Nov 15 at 12:13
MathematicianP
3315
asked Nov 15 at 12:13
MathematicianP
3315
asked Nov 15 at 12:13
MathematicianP
3315
asked Nov 15 at 12:13
MathematicianP
3315
asked Nov 15 at 12:13
MathematicianP
3315
3315
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
You are right, but for the wrong reason. Being closed is a topological property because it depends only upon the topology induced by the metric, not by the metric itself. For instance, being bounded is not a topological property, because a set may be bounded with respect to a metric and unbounded with respect to another equivalent metric.
On the other hand, it is false that if a set is closed with respect to a metric then it will automatically be closed with respect to any other metric. For instance, in $mathbb R$, $[0,1)$ is closed with respect to the discrete metric, but not with respect to the usual one.
answered Nov 15 at 12:17
José Carlos Santos
142k20111207
1
but why does it depend only on the topology induced by the metric?
– MathematicianP
Nov 15 at 12:24
Because being closed means that its complement is open and the set of open subsets is the topology.
– José Carlos Santos
Nov 15 at 12:27
Thank you for your clarity!!
– MathematicianP
Nov 15 at 12:37
I'm glad I could help.
– José Carlos Santos
Nov 15 at 12:38
Is it therefore true that being closed and bounded is not a topological property?
– MathematicianP
Nov 15 at 12:43
|
show 1 more comment
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
You are right, but for the wrong reason. Being closed is a topological property because it depends only upon the topology induced by the metric, not by the metric itself. For instance, being bounded is not a topological property, because a set may be bounded with respect to a metric and unbounded with respect to another equivalent metric.
On the other hand, it is false that if a set is closed with respect to a metric then it will automatically be closed with respect to any other metric. For instance, in $mathbb R$, $[0,1)$ is closed with respect to the discrete metric, but not with respect to the usual one.
answered Nov 15 at 12:17
José Carlos Santos
142k20111207
1
but why does it depend only on the topology induced by the metric?
– MathematicianP
Nov 15 at 12:24
Because being closed means that its complement is open and the set of open subsets is the topology.
– José Carlos Santos
Nov 15 at 12:27
Thank you for your clarity!!
– MathematicianP
Nov 15 at 12:37
I'm glad I could help.
– José Carlos Santos
Nov 15 at 12:38
Is it therefore true that being closed and bounded is not a topological property?
– MathematicianP
Nov 15 at 12:43
|
show 1 more comment
up vote
1
down vote
accepted
You are right, but for the wrong reason. Being closed is a topological property because it depends only upon the topology induced by the metric, not by the metric itself. For instance, being bounded is not a topological property, because a set may be bounded with respect to a metric and unbounded with respect to another equivalent metric.
On the other hand, it is false that if a set is closed with respect to a metric then it will automatically be closed with respect to any other metric. For instance, in $mathbb R$, $[0,1)$ is closed with respect to the discrete metric, but not with respect to the usual one.
answered Nov 15 at 12:17
José Carlos Santos
142k20111207
1
but why does it depend only on the topology induced by the metric?
– MathematicianP
Nov 15 at 12:24
Because being closed means that its complement is open and the set of open subsets is the topology.
– José Carlos Santos
Nov 15 at 12:27
Thank you for your clarity!!
– MathematicianP
Nov 15 at 12:37
I'm glad I could help.
– José Carlos Santos
Nov 15 at 12:38
Is it therefore true that being closed and bounded is not a topological property?
– MathematicianP
Nov 15 at 12:43
|
show 1 more comment
up vote
1
down vote
accepted
up vote
1
down vote
accepted
You are right, but for the wrong reason. Being closed is a topological property because it depends only upon the topology induced by the metric, not by the metric itself. For instance, being bounded is not a topological property, because a set may be bounded with respect to a metric and unbounded with respect to another equivalent metric.
On the other hand, it is false that if a set is closed with respect to a metric then it will automatically be closed with respect to any other metric. For instance, in $mathbb R$, $[0,1)$ is closed with respect to the discrete metric, but not with respect to the usual one.
answered Nov 15 at 12:17
José Carlos Santos
142k20111207
You are right, but for the wrong reason. Being closed is a topological property because it depends only upon the topology induced by the metric, not by the metric itself. For instance, being bounded is not a topological property, because a set may be bounded with respect to a metric and unbounded with respect to another equivalent metric.
On the other hand, it is false that if a set is closed with respect to a metric then it will automatically be closed with respect to any other metric. For instance, in $mathbb R$, $[0,1)$ is closed with respect to the discrete metric, but not with respect to the usual one.
answered Nov 15 at 12:17
José Carlos Santos
142k20111207
answered Nov 15 at 12:17
José Carlos Santos
142k20111207
answered Nov 15 at 12:17
José Carlos Santos
142k20111207
answered Nov 15 at 12:17
José Carlos Santos
142k20111207
142k20111207
1
but why does it depend only on the topology induced by the metric?
– MathematicianP
Nov 15 at 12:24
Because being closed means that its complement is open and the set of open subsets is the topology.
– José Carlos Santos
Nov 15 at 12:27
Thank you for your clarity!!
– MathematicianP
Nov 15 at 12:37
I'm glad I could help.
– José Carlos Santos
Nov 15 at 12:38
Is it therefore true that being closed and bounded is not a topological property?
– MathematicianP
Nov 15 at 12:43
|
show 1 more comment
1
but why does it depend only on the topology induced by the metric?
– MathematicianP
Nov 15 at 12:24
Because being closed means that its complement is open and the set of open subsets is the topology.
– José Carlos Santos
Nov 15 at 12:27
Thank you for your clarity!!
– MathematicianP
Nov 15 at 12:37
I'm glad I could help.
– José Carlos Santos
Nov 15 at 12:38
Is it therefore true that being closed and bounded is not a topological property?
– MathematicianP
Nov 15 at 12:43
1
1
but why does it depend only on the topology induced by the metric?
– MathematicianP
Nov 15 at 12:24
but why does it depend only on the topology induced by the metric?
– MathematicianP
Nov 15 at 12:24
Because being closed means that its complement is open and the set of open subsets is the topology.
– José Carlos Santos
Nov 15 at 12:27
Because being closed means that its complement is open and the set of open subsets is the topology.
– José Carlos Santos
Nov 15 at 12:27
Thank you for your clarity!!
– MathematicianP
Nov 15 at 12:37
Thank you for your clarity!!
– MathematicianP
Nov 15 at 12:37
I'm glad I could help.
– José Carlos Santos
Nov 15 at 12:38
I'm glad I could help.
– José Carlos Santos
Nov 15 at 12:38
Is it therefore true that being closed and bounded is not a topological property?
– MathematicianP
Nov 15 at 12:43
Is it therefore true that being closed and bounded is not a topological property?
– MathematicianP
Nov 15 at 12:43
|
show 1 more comment
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