Standard definition of subnets. Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Different definitions of subnetRelation between convergence class and convergence spaceMoore–Smith convergence; Universal netsSubnets and finer filtersEvery net has an ultranet as subnet: direct proofConverse of closed graph theorem in general topological spaceHelp me understand this passage from “General Topology” by J. KelleySequential compactness implies compactness: what is wrong with this argument?How to build a subnet out of these subnets?Understanding product topologyA net has a limit if and only if all of its subnets have limits (without the use of Cauchy nets)
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Standard definition of subnets.
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Different definitions of subnetRelation between convergence class and convergence spaceMoore–Smith convergence; Universal netsSubnets and finer filtersEvery net has an ultranet as subnet: direct proofConverse of closed graph theorem in general topological spaceHelp me understand this passage from “General Topology” by J. KelleySequential compactness implies compactness: what is wrong with this argument?How to build a subnet out of these subnets?Understanding product topologyA net has a limit if and only if all of its subnets have limits (without the use of Cauchy nets)
$begingroup$
Reading Willard's General Topology, I found the following definition of a subnet of a net $f:Dto X$:
$g:Eto X$ is a subnet of $f$ if there exists an increasing $varphi:Eto D$ such that $g=fcircvarphi$ and $forall alphain D, exists beta in E, varphi(beta)gealpha$;
while, reading Kelley's General Topology, I found the following definition of a subnet of net $f:Dto X$:
$g:Eto X$ is a subnet of $f$ if there exists $varphi:Eto D$ such that $g=fcircvarphi$ and$forall alphain D, exists beta_0 in E, forallbeta gebeta_0, varphi(beta)gealpha.$
Clearly, Willard's definition is strictly stronger than Kelley's definition (i.e. every subnet according Willard is a subnet according Kelley but in general not the converse). Obviously we can use both definitions to obtain the standard theorems about subnets we were looking for, so it seems just a matter of tastes which definition we decide to use. However:
In the literature, which one of the two definitions has become standard? Willard's or Kelley's?
general-topology nets
asked Mar 26 at 7:28
BobBob
1,7051725
$endgroup$
add a comment |
$begingroup$
Reading Willard's General Topology, I found the following definition of a subnet of a net $f:Dto X$:
$g:Eto X$ is a subnet of $f$ if there exists an increasing $varphi:Eto D$ such that $g=fcircvarphi$ and $forall alphain D, exists beta in E, varphi(beta)gealpha$;
while, reading Kelley's General Topology, I found the following definition of a subnet of net $f:Dto X$:
$g:Eto X$ is a subnet of $f$ if there exists $varphi:Eto D$ such that $g=fcircvarphi$ and$forall alphain D, exists beta_0 in E, forallbeta gebeta_0, varphi(beta)gealpha.$
Clearly, Willard's definition is strictly stronger than Kelley's definition (i.e. every subnet according Willard is a subnet according Kelley but in general not the converse). Obviously we can use both definitions to obtain the standard theorems about subnets we were looking for, so it seems just a matter of tastes which definition we decide to use. However:
In the literature, which one of the two definitions has become standard? Willard's or Kelley's?
general-topology nets
asked Mar 26 at 7:28
BobBob
1,7051725
$endgroup$
$begingroup$
You may be interested in this question
$endgroup$
– MPW
Mar 26 at 7:43
$begingroup$
Thx for the comment... at this point I see that I have problems in searching topics in this site...
$endgroup$
– Bob
Mar 26 at 7:55
$begingroup$
For what it’s worth, so do I. I Googled it instead, found this question there. I think it’s very difficult to search this site.
$endgroup$
– MPW
Mar 26 at 7:57
add a comment |
$begingroup$
Reading Willard's General Topology, I found the following definition of a subnet of a net $f:Dto X$:
$g:Eto X$ is a subnet of $f$ if there exists an increasing $varphi:Eto D$ such that $g=fcircvarphi$ and $forall alphain D, exists beta in E, varphi(beta)gealpha$;
while, reading Kelley's General Topology, I found the following definition of a subnet of net $f:Dto X$:
$g:Eto X$ is a subnet of $f$ if there exists $varphi:Eto D$ such that $g=fcircvarphi$ and$forall alphain D, exists beta_0 in E, forallbeta gebeta_0, varphi(beta)gealpha.$
Clearly, Willard's definition is strictly stronger than Kelley's definition (i.e. every subnet according Willard is a subnet according Kelley but in general not the converse). Obviously we can use both definitions to obtain the standard theorems about subnets we were looking for, so it seems just a matter of tastes which definition we decide to use. However:
In the literature, which one of the two definitions has become standard? Willard's or Kelley's?
general-topology nets
asked Mar 26 at 7:28
BobBob
1,7051725
$endgroup$
Reading Willard's General Topology, I found the following definition of a subnet of a net $f:Dto X$:
$g:Eto X$ is a subnet of $f$ if there exists an increasing $varphi:Eto D$ such that $g=fcircvarphi$ and $forall alphain D, exists beta in E, varphi(beta)gealpha$;
while, reading Kelley's General Topology, I found the following definition of a subnet of net $f:Dto X$:
$g:Eto X$ is a subnet of $f$ if there exists $varphi:Eto D$ such that $g=fcircvarphi$ and$forall alphain D, exists beta_0 in E, forallbeta gebeta_0, varphi(beta)gealpha.$
Clearly, Willard's definition is strictly stronger than Kelley's definition (i.e. every subnet according Willard is a subnet according Kelley but in general not the converse). Obviously we can use both definitions to obtain the standard theorems about subnets we were looking for, so it seems just a matter of tastes which definition we decide to use. However:
In the literature, which one of the two definitions has become standard? Willard's or Kelley's?
general-topology nets
general-topology nets
asked Mar 26 at 7:28
BobBob
1,7051725
asked Mar 26 at 7:28
BobBob
1,7051725
edited Mar 26 at 12:34
Bob
asked Mar 26 at 7:28
BobBob
1,7051725
asked Mar 26 at 7:28
BobBob
1,7051725
asked Mar 26 at 7:28
BobBob
1,7051725
1,7051725
$begingroup$
You may be interested in this question
$endgroup$
– MPW
Mar 26 at 7:43
$begingroup$
Thx for the comment... at this point I see that I have problems in searching topics in this site...
$endgroup$
– Bob
Mar 26 at 7:55
$begingroup$
For what it’s worth, so do I. I Googled it instead, found this question there. I think it’s very difficult to search this site.
$endgroup$
– MPW
Mar 26 at 7:57
add a comment |
$begingroup$
You may be interested in this question
$endgroup$
– MPW
Mar 26 at 7:43
$begingroup$
Thx for the comment... at this point I see that I have problems in searching topics in this site...
$endgroup$
– Bob
Mar 26 at 7:55
$begingroup$
For what it’s worth, so do I. I Googled it instead, found this question there. I think it’s very difficult to search this site.
$endgroup$
– MPW
Mar 26 at 7:57
$begingroup$
You may be interested in this question
$endgroup$
– MPW
Mar 26 at 7:43
$begingroup$
You may be interested in this question
$endgroup$
– MPW
Mar 26 at 7:43
$begingroup$
Thx for the comment... at this point I see that I have problems in searching topics in this site...
$endgroup$
– Bob
Mar 26 at 7:55
$begingroup$
Thx for the comment... at this point I see that I have problems in searching topics in this site...
$endgroup$
– Bob
Mar 26 at 7:55
$begingroup$
For what it’s worth, so do I. I Googled it instead, found this question there. I think it’s very difficult to search this site.
$endgroup$
– MPW
Mar 26 at 7:57
$begingroup$
For what it’s worth, so do I. I Googled it instead, found this question there. I think it’s very difficult to search this site.
$endgroup$
– MPW
Mar 26 at 7:57
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This is indeed mostly a matter of taste; Willard's definition is more in the style of subsequences (maybe easier to visualise), while Kelley's is more general (so easier to fulfil, which can be easier to construct in proofs e.g.) and more in the generality spirit of directed sets, as it were. A subnet in Kelley's sense need not be one in Willard's sense, but the difference is quiet subtle. In terms of translating to filters a third notion (AA-subnet) is also sometimes used. See the discussion in this thread e.g. Most papers I've seen tend to use filters or use Kelley's definition (but I haven't done a count of popularity).
answered Mar 26 at 7:58
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
A subnet of the net f from D into X is f restricted to a cofinal subset of D. Is not that another equivalent definition?
$endgroup$
– William Elliot
Mar 26 at 8:12
$begingroup$
@WilliamElliot no, that's much too restrictive; you don't get the theorem that a space is compact iff every net has a convergent subnet, which is one of the main reasons for defining subnet in the first place. A sequence in your proposal only subnets that are subsequences while there are quite a few compact but not sequentially compact spaces...
$endgroup$
– Henno Brandsma
Mar 26 at 13:11
add a comment |
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1 Answer
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1 Answer
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$begingroup$
This is indeed mostly a matter of taste; Willard's definition is more in the style of subsequences (maybe easier to visualise), while Kelley's is more general (so easier to fulfil, which can be easier to construct in proofs e.g.) and more in the generality spirit of directed sets, as it were. A subnet in Kelley's sense need not be one in Willard's sense, but the difference is quiet subtle. In terms of translating to filters a third notion (AA-subnet) is also sometimes used. See the discussion in this thread e.g. Most papers I've seen tend to use filters or use Kelley's definition (but I haven't done a count of popularity).
answered Mar 26 at 7:58
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
A subnet of the net f from D into X is f restricted to a cofinal subset of D. Is not that another equivalent definition?
$endgroup$
– William Elliot
Mar 26 at 8:12
$begingroup$
@WilliamElliot no, that's much too restrictive; you don't get the theorem that a space is compact iff every net has a convergent subnet, which is one of the main reasons for defining subnet in the first place. A sequence in your proposal only subnets that are subsequences while there are quite a few compact but not sequentially compact spaces...
$endgroup$
– Henno Brandsma
Mar 26 at 13:11
add a comment |
$begingroup$
This is indeed mostly a matter of taste; Willard's definition is more in the style of subsequences (maybe easier to visualise), while Kelley's is more general (so easier to fulfil, which can be easier to construct in proofs e.g.) and more in the generality spirit of directed sets, as it were. A subnet in Kelley's sense need not be one in Willard's sense, but the difference is quiet subtle. In terms of translating to filters a third notion (AA-subnet) is also sometimes used. See the discussion in this thread e.g. Most papers I've seen tend to use filters or use Kelley's definition (but I haven't done a count of popularity).
answered Mar 26 at 7:58
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
A subnet of the net f from D into X is f restricted to a cofinal subset of D. Is not that another equivalent definition?
$endgroup$
– William Elliot
Mar 26 at 8:12
$begingroup$
@WilliamElliot no, that's much too restrictive; you don't get the theorem that a space is compact iff every net has a convergent subnet, which is one of the main reasons for defining subnet in the first place. A sequence in your proposal only subnets that are subsequences while there are quite a few compact but not sequentially compact spaces...
$endgroup$
– Henno Brandsma
Mar 26 at 13:11
add a comment |
$begingroup$
This is indeed mostly a matter of taste; Willard's definition is more in the style of subsequences (maybe easier to visualise), while Kelley's is more general (so easier to fulfil, which can be easier to construct in proofs e.g.) and more in the generality spirit of directed sets, as it were. A subnet in Kelley's sense need not be one in Willard's sense, but the difference is quiet subtle. In terms of translating to filters a third notion (AA-subnet) is also sometimes used. See the discussion in this thread e.g. Most papers I've seen tend to use filters or use Kelley's definition (but I haven't done a count of popularity).
answered Mar 26 at 7:58
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
This is indeed mostly a matter of taste; Willard's definition is more in the style of subsequences (maybe easier to visualise), while Kelley's is more general (so easier to fulfil, which can be easier to construct in proofs e.g.) and more in the generality spirit of directed sets, as it were. A subnet in Kelley's sense need not be one in Willard's sense, but the difference is quiet subtle. In terms of translating to filters a third notion (AA-subnet) is also sometimes used. See the discussion in this thread e.g. Most papers I've seen tend to use filters or use Kelley's definition (but I haven't done a count of popularity).
answered Mar 26 at 7:58
Henno BrandsmaHenno Brandsma
116k349127
answered Mar 26 at 7:58
Henno BrandsmaHenno Brandsma
116k349127
answered Mar 26 at 7:58
Henno BrandsmaHenno Brandsma
116k349127
answered Mar 26 at 7:58
Henno BrandsmaHenno Brandsma
116k349127
116k349127
$begingroup$
A subnet of the net f from D into X is f restricted to a cofinal subset of D. Is not that another equivalent definition?
$endgroup$
– William Elliot
Mar 26 at 8:12
$begingroup$
@WilliamElliot no, that's much too restrictive; you don't get the theorem that a space is compact iff every net has a convergent subnet, which is one of the main reasons for defining subnet in the first place. A sequence in your proposal only subnets that are subsequences while there are quite a few compact but not sequentially compact spaces...
$endgroup$
– Henno Brandsma
Mar 26 at 13:11
add a comment |
$begingroup$
A subnet of the net f from D into X is f restricted to a cofinal subset of D. Is not that another equivalent definition?
$endgroup$
– William Elliot
Mar 26 at 8:12
$begingroup$
@WilliamElliot no, that's much too restrictive; you don't get the theorem that a space is compact iff every net has a convergent subnet, which is one of the main reasons for defining subnet in the first place. A sequence in your proposal only subnets that are subsequences while there are quite a few compact but not sequentially compact spaces...
$endgroup$
– Henno Brandsma
Mar 26 at 13:11
$begingroup$
A subnet of the net f from D into X is f restricted to a cofinal subset of D. Is not that another equivalent definition?
$endgroup$
– William Elliot
Mar 26 at 8:12
$begingroup$
A subnet of the net f from D into X is f restricted to a cofinal subset of D. Is not that another equivalent definition?
$endgroup$
– William Elliot
Mar 26 at 8:12
$begingroup$
@WilliamElliot no, that's much too restrictive; you don't get the theorem that a space is compact iff every net has a convergent subnet, which is one of the main reasons for defining subnet in the first place. A sequence in your proposal only subnets that are subsequences while there are quite a few compact but not sequentially compact spaces...
$endgroup$
– Henno Brandsma
Mar 26 at 13:11
$begingroup$
@WilliamElliot no, that's much too restrictive; you don't get the theorem that a space is compact iff every net has a convergent subnet, which is one of the main reasons for defining subnet in the first place. A sequence in your proposal only subnets that are subsequences while there are quite a few compact but not sequentially compact spaces...
$endgroup$
– Henno Brandsma
Mar 26 at 13:11
add a comment |
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$begingroup$
You may be interested in this question
$endgroup$
– MPW
Mar 26 at 7:43
$begingroup$
Thx for the comment... at this point I see that I have problems in searching topics in this site...
$endgroup$
– Bob
Mar 26 at 7:55
$begingroup$
For what it’s worth, so do I. I Googled it instead, found this question there. I think it’s very difficult to search this site.
$endgroup$
– MPW
Mar 26 at 7:57