Difference between $R^infty$ and $R^omega$
4
$begingroup$
I know $R^omega$ is the set of functions from $omega$ to $R$. I would think $R^infty$ as the limit of $R^n$, but isn't that $R^omega$?
The seem to be used differently, but I can't tell exactly how.
general-topology elementary-set-theory notation
edited Dec 23 '13 at 17:06
lhf
167k11172403
asked Dec 23 '13 at 16:53
Thomas AhleThomas Ahle
1,5091323
$endgroup$
add a comment |
4
$begingroup$
I know $R^omega$ is the set of functions from $omega$ to $R$. I would think $R^infty$ as the limit of $R^n$, but isn't that $R^omega$?
The seem to be used differently, but I can't tell exactly how.
general-topology elementary-set-theory notation
edited Dec 23 '13 at 17:06
lhf
167k11172403
asked Dec 23 '13 at 16:53
Thomas AhleThomas Ahle
1,5091323
$endgroup$
2
$begingroup$
Notations mean what their users want. Without context, it is impossible to decide what either of these notatins you mention mean. (Limit in what sense?)
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:54
2
$begingroup$
This will manifest itself in people answering completely different things, due to their familiarity with different conventions. It does not helpthat each answer appears to think that their answer is authoritative :-(
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:57
add a comment |
4
4
4
3
$begingroup$
I know $R^omega$ is the set of functions from $omega$ to $R$. I would think $R^infty$ as the limit of $R^n$, but isn't that $R^omega$?
The seem to be used differently, but I can't tell exactly how.
general-topology elementary-set-theory notation
edited Dec 23 '13 at 17:06
lhf
167k11172403
asked Dec 23 '13 at 16:53
Thomas AhleThomas Ahle
1,5091323
$endgroup$
I know $R^omega$ is the set of functions from $omega$ to $R$. I would think $R^infty$ as the limit of $R^n$, but isn't that $R^omega$?
The seem to be used differently, but I can't tell exactly how.
general-topology elementary-set-theory notation
general-topology elementary-set-theory notation
edited Dec 23 '13 at 17:06
lhf
167k11172403
asked Dec 23 '13 at 16:53
Thomas AhleThomas Ahle
1,5091323
edited Dec 23 '13 at 17:06
lhf
167k11172403
asked Dec 23 '13 at 16:53
Thomas AhleThomas Ahle
1,5091323
edited Dec 23 '13 at 17:06
lhf
167k11172403
edited Dec 23 '13 at 17:06
lhf
167k11172403
edited Dec 23 '13 at 17:06
lhf
167k11172403
167k11172403
asked Dec 23 '13 at 16:53
Thomas AhleThomas Ahle
1,5091323
asked Dec 23 '13 at 16:53
Thomas AhleThomas Ahle
1,5091323
asked Dec 23 '13 at 16:53
Thomas AhleThomas Ahle
1,5091323
1,5091323
2
$begingroup$
Notations mean what their users want. Without context, it is impossible to decide what either of these notatins you mention mean. (Limit in what sense?)
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:54
2
$begingroup$
This will manifest itself in people answering completely different things, due to their familiarity with different conventions. It does not helpthat each answer appears to think that their answer is authoritative :-(
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:57
add a comment |
2
$begingroup$
Notations mean what their users want. Without context, it is impossible to decide what either of these notatins you mention mean. (Limit in what sense?)
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:54
2
$begingroup$
This will manifest itself in people answering completely different things, due to their familiarity with different conventions. It does not helpthat each answer appears to think that their answer is authoritative :-(
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:57
2
2
$begingroup$
Notations mean what their users want. Without context, it is impossible to decide what either of these notatins you mention mean. (Limit in what sense?)
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:54
$begingroup$
Notations mean what their users want. Without context, it is impossible to decide what either of these notatins you mention mean. (Limit in what sense?)
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:54
2
2
$begingroup$
This will manifest itself in people answering completely different things, due to their familiarity with different conventions. It does not helpthat each answer appears to think that their answer is authoritative :-(
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:57
$begingroup$
This will manifest itself in people answering completely different things, due to their familiarity with different conventions. It does not helpthat each answer appears to think that their answer is authoritative :-(
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:57
add a comment |
1 Answer
1
active
oldest
votes
10
$begingroup$
In a context where one makes a distiction between $R^infty$ and $R^omega$,
$R^infty$ denotes the set of sequences with finite support whereas $R^omega$ denotes the set of unrestricted sequences.
In this context, $R^infty$ is the limit of $R^n$ when $n to infty$, in the sense that $R^infty = bigcup_{n=0}^{infty} R^n$, with the convention that $R^n$ are all seen as subsets of $R^omega$.
answered Dec 23 '13 at 16:56
lhflhf
167k11172403
$endgroup$
$begingroup$
Usually? ${}{}{}$
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:56
1
$begingroup$
@ThomasAhle, see my edited answer.
$endgroup$
– lhf
Dec 23 '13 at 17:03
$begingroup$
To extend the original question: What are default topologies on $R^∞$ and $R^ω$? I would think that we take product topology on $R^ω$ and limit topology on $R^∞$ (in the sense of direct limit in category of topological spaces, if I remeber in this case it corresponds to subspace topology of box product).
$endgroup$
– user87690
Dec 23 '13 at 17:27
$begingroup$
@user87690, perhaps you could ask a separate question about topology.
$endgroup$
– lhf
Dec 23 '13 at 17:38
$begingroup$
@lhf: Ok. Done.
$endgroup$
– user87690
Dec 23 '13 at 20:03
add a comment |
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1 Answer
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oldest
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
10
$begingroup$
In a context where one makes a distiction between $R^infty$ and $R^omega$,
$R^infty$ denotes the set of sequences with finite support whereas $R^omega$ denotes the set of unrestricted sequences.
In this context, $R^infty$ is the limit of $R^n$ when $n to infty$, in the sense that $R^infty = bigcup_{n=0}^{infty} R^n$, with the convention that $R^n$ are all seen as subsets of $R^omega$.
answered Dec 23 '13 at 16:56
lhflhf
167k11172403
$endgroup$
$begingroup$
Usually? ${}{}{}$
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:56
1
$begingroup$
@ThomasAhle, see my edited answer.
$endgroup$
– lhf
Dec 23 '13 at 17:03
$begingroup$
To extend the original question: What are default topologies on $R^∞$ and $R^ω$? I would think that we take product topology on $R^ω$ and limit topology on $R^∞$ (in the sense of direct limit in category of topological spaces, if I remeber in this case it corresponds to subspace topology of box product).
$endgroup$
– user87690
Dec 23 '13 at 17:27
$begingroup$
@user87690, perhaps you could ask a separate question about topology.
$endgroup$
– lhf
Dec 23 '13 at 17:38
$begingroup$
@lhf: Ok. Done.
$endgroup$
– user87690
Dec 23 '13 at 20:03
add a comment |
10
$begingroup$
In a context where one makes a distiction between $R^infty$ and $R^omega$,
$R^infty$ denotes the set of sequences with finite support whereas $R^omega$ denotes the set of unrestricted sequences.
In this context, $R^infty$ is the limit of $R^n$ when $n to infty$, in the sense that $R^infty = bigcup_{n=0}^{infty} R^n$, with the convention that $R^n$ are all seen as subsets of $R^omega$.
answered Dec 23 '13 at 16:56
lhflhf
167k11172403
$endgroup$
$begingroup$
Usually? ${}{}{}$
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:56
1
$begingroup$
@ThomasAhle, see my edited answer.
$endgroup$
– lhf
Dec 23 '13 at 17:03
$begingroup$
To extend the original question: What are default topologies on $R^∞$ and $R^ω$? I would think that we take product topology on $R^ω$ and limit topology on $R^∞$ (in the sense of direct limit in category of topological spaces, if I remeber in this case it corresponds to subspace topology of box product).
$endgroup$
– user87690
Dec 23 '13 at 17:27
$begingroup$
@user87690, perhaps you could ask a separate question about topology.
$endgroup$
– lhf
Dec 23 '13 at 17:38
$begingroup$
@lhf: Ok. Done.
$endgroup$
– user87690
Dec 23 '13 at 20:03
add a comment |
10
10
10
$begingroup$
In a context where one makes a distiction between $R^infty$ and $R^omega$,
$R^infty$ denotes the set of sequences with finite support whereas $R^omega$ denotes the set of unrestricted sequences.
In this context, $R^infty$ is the limit of $R^n$ when $n to infty$, in the sense that $R^infty = bigcup_{n=0}^{infty} R^n$, with the convention that $R^n$ are all seen as subsets of $R^omega$.
answered Dec 23 '13 at 16:56
lhflhf
167k11172403
$endgroup$
In a context where one makes a distiction between $R^infty$ and $R^omega$,
$R^infty$ denotes the set of sequences with finite support whereas $R^omega$ denotes the set of unrestricted sequences.
In this context, $R^infty$ is the limit of $R^n$ when $n to infty$, in the sense that $R^infty = bigcup_{n=0}^{infty} R^n$, with the convention that $R^n$ are all seen as subsets of $R^omega$.
answered Dec 23 '13 at 16:56
lhflhf
167k11172403
edited Dec 23 '13 at 17:02
answered Dec 23 '13 at 16:56
lhflhf
167k11172403
answered Dec 23 '13 at 16:56
lhflhf
167k11172403
answered Dec 23 '13 at 16:56
lhflhf
167k11172403
167k11172403
$begingroup$
Usually? ${}{}{}$
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:56
1
$begingroup$
@ThomasAhle, see my edited answer.
$endgroup$
– lhf
Dec 23 '13 at 17:03
$begingroup$
To extend the original question: What are default topologies on $R^∞$ and $R^ω$? I would think that we take product topology on $R^ω$ and limit topology on $R^∞$ (in the sense of direct limit in category of topological spaces, if I remeber in this case it corresponds to subspace topology of box product).
$endgroup$
– user87690
Dec 23 '13 at 17:27
$begingroup$
@user87690, perhaps you could ask a separate question about topology.
$endgroup$
– lhf
Dec 23 '13 at 17:38
$begingroup$
@lhf: Ok. Done.
$endgroup$
– user87690
Dec 23 '13 at 20:03
add a comment |
$begingroup$
Usually? ${}{}{}$
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:56
1
$begingroup$
@ThomasAhle, see my edited answer.
$endgroup$
– lhf
Dec 23 '13 at 17:03
$begingroup$
To extend the original question: What are default topologies on $R^∞$ and $R^ω$? I would think that we take product topology on $R^ω$ and limit topology on $R^∞$ (in the sense of direct limit in category of topological spaces, if I remeber in this case it corresponds to subspace topology of box product).
$endgroup$
– user87690
Dec 23 '13 at 17:27
$begingroup$
@user87690, perhaps you could ask a separate question about topology.
$endgroup$
– lhf
Dec 23 '13 at 17:38
$begingroup$
@lhf: Ok. Done.
$endgroup$
– user87690
Dec 23 '13 at 20:03
$begingroup$
Usually? ${}{}{}$
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:56
$begingroup$
Usually? ${}{}{}$
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:56
1
1
$begingroup$
@ThomasAhle, see my edited answer.
$endgroup$
– lhf
Dec 23 '13 at 17:03
$begingroup$
@ThomasAhle, see my edited answer.
$endgroup$
– lhf
Dec 23 '13 at 17:03
$begingroup$
To extend the original question: What are default topologies on $R^∞$ and $R^ω$? I would think that we take product topology on $R^ω$ and limit topology on $R^∞$ (in the sense of direct limit in category of topological spaces, if I remeber in this case it corresponds to subspace topology of box product).
$endgroup$
– user87690
Dec 23 '13 at 17:27
$begingroup$
To extend the original question: What are default topologies on $R^∞$ and $R^ω$? I would think that we take product topology on $R^ω$ and limit topology on $R^∞$ (in the sense of direct limit in category of topological spaces, if I remeber in this case it corresponds to subspace topology of box product).
$endgroup$
– user87690
Dec 23 '13 at 17:27
$begingroup$
@user87690, perhaps you could ask a separate question about topology.
$endgroup$
– lhf
Dec 23 '13 at 17:38
$begingroup$
@user87690, perhaps you could ask a separate question about topology.
$endgroup$
– lhf
Dec 23 '13 at 17:38
$begingroup$
@lhf: Ok. Done.
$endgroup$
– user87690
Dec 23 '13 at 20:03
$begingroup$
@lhf: Ok. Done.
$endgroup$
– user87690
Dec 23 '13 at 20:03
add a comment |
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2
$begingroup$
Notations mean what their users want. Without context, it is impossible to decide what either of these notatins you mention mean. (Limit in what sense?)
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:54
2
$begingroup$
This will manifest itself in people answering completely different things, due to their familiarity with different conventions. It does not helpthat each answer appears to think that their answer is authoritative :-(
$endgroup$
– Mariano Suárez-Álvarez
Dec 23 '13 at 16:57