Preservation of compactness under equivalent metrics
$begingroup$
Let $X$ be a compact metric space under a certain metric $d_1$.
Let $d_2$ be a metric equivalent to $d_1$. Is it true that $(X,d_2)$ is a compact metric space?
If so how do I go about proving it?
general-topology metric-spaces compactness
edited Dec 30 '18 at 11:05
Javier
2,01621133
asked Apr 16 '18 at 17:54
Anand RaviAnand Ravi
62
$endgroup$
add a comment |
$begingroup$
Let $X$ be a compact metric space under a certain metric $d_1$.
Let $d_2$ be a metric equivalent to $d_1$. Is it true that $(X,d_2)$ is a compact metric space?
If so how do I go about proving it?
general-topology metric-spaces compactness
edited Dec 30 '18 at 11:05
Javier
2,01621133
asked Apr 16 '18 at 17:54
Anand RaviAnand Ravi
62
$endgroup$
$begingroup$
Compactness is a property of the topology. Equivalent metrics generate the same topology.
$endgroup$
– DanielWainfleet
Apr 18 '18 at 5:27
add a comment |
$begingroup$
Let $X$ be a compact metric space under a certain metric $d_1$.
Let $d_2$ be a metric equivalent to $d_1$. Is it true that $(X,d_2)$ is a compact metric space?
If so how do I go about proving it?
general-topology metric-spaces compactness
edited Dec 30 '18 at 11:05
Javier
2,01621133
asked Apr 16 '18 at 17:54
Anand RaviAnand Ravi
62
$endgroup$
Let $X$ be a compact metric space under a certain metric $d_1$.
Let $d_2$ be a metric equivalent to $d_1$. Is it true that $(X,d_2)$ is a compact metric space?
If so how do I go about proving it?
general-topology metric-spaces compactness
general-topology metric-spaces compactness
edited Dec 30 '18 at 11:05
Javier
2,01621133
asked Apr 16 '18 at 17:54
Anand RaviAnand Ravi
62
edited Dec 30 '18 at 11:05
Javier
2,01621133
asked Apr 16 '18 at 17:54
Anand RaviAnand Ravi
62
edited Dec 30 '18 at 11:05
Javier
2,01621133
edited Dec 30 '18 at 11:05
Javier
2,01621133
edited Dec 30 '18 at 11:05
Javier
2,01621133
2,01621133
asked Apr 16 '18 at 17:54
Anand RaviAnand Ravi
62
asked Apr 16 '18 at 17:54
Anand RaviAnand Ravi
62
asked Apr 16 '18 at 17:54
Anand RaviAnand Ravi
62
62
$begingroup$
Compactness is a property of the topology. Equivalent metrics generate the same topology.
$endgroup$
– DanielWainfleet
Apr 18 '18 at 5:27
add a comment |
$begingroup$
Compactness is a property of the topology. Equivalent metrics generate the same topology.
$endgroup$
– DanielWainfleet
Apr 18 '18 at 5:27
$begingroup$
Compactness is a property of the topology. Equivalent metrics generate the same topology.
$endgroup$
– DanielWainfleet
Apr 18 '18 at 5:27
$begingroup$
Compactness is a property of the topology. Equivalent metrics generate the same topology.
$endgroup$
– DanielWainfleet
Apr 18 '18 at 5:27
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes, it is true, since asserting that the metrics are equivalent means that the topologies are the same.
answered Apr 16 '18 at 17:58
José Carlos SantosJosé Carlos Santos
154k22123226
$endgroup$
add a comment |
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$begingroup$
Yes, it is true, since asserting that the metrics are equivalent means that the topologies are the same.
answered Apr 16 '18 at 17:58
José Carlos SantosJosé Carlos Santos
154k22123226
$endgroup$
add a comment |
$begingroup$
Yes, it is true, since asserting that the metrics are equivalent means that the topologies are the same.
answered Apr 16 '18 at 17:58
José Carlos SantosJosé Carlos Santos
154k22123226
$endgroup$
add a comment |
$begingroup$
Yes, it is true, since asserting that the metrics are equivalent means that the topologies are the same.
answered Apr 16 '18 at 17:58
José Carlos SantosJosé Carlos Santos
154k22123226
$endgroup$
Yes, it is true, since asserting that the metrics are equivalent means that the topologies are the same.
answered Apr 16 '18 at 17:58
José Carlos SantosJosé Carlos Santos
154k22123226
answered Apr 16 '18 at 17:58
José Carlos SantosJosé Carlos Santos
154k22123226
answered Apr 16 '18 at 17:58
José Carlos SantosJosé Carlos Santos
154k22123226
answered Apr 16 '18 at 17:58
José Carlos SantosJosé Carlos Santos
154k22123226
154k22123226
add a comment |
add a comment |
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$begingroup$
Compactness is a property of the topology. Equivalent metrics generate the same topology.
$endgroup$
– DanielWainfleet
Apr 18 '18 at 5:27