Mixing
Is there a notation for “open subset”
1
$begingroup$
I very often have to write something like:
$exists U,Vsubseteq M$ where $U,V$ are open, but there's no short hand for it. On my written notes, I do tend to write something like:
$exists U,Vmathop{subseteq}_text{open}M$ is there a common short hand though?
Example of a short hand that exists
$newcommand{bigudot}{mathop{bigcupmkern-14mucdotmkern5mu}}$
$newcommand{udot}{cupmkern-11.5mucdotmkern5mu}$
It is established already that $bigudot$ means "union of sets that are pairwise disjoint", so if I write:
$forall Aexistsmathcal{A}:Asubseteqbigudotmathcal{A}$ - or something - it is clear from the context that $mathcal{A}$ is a family of pairwise disjoint sets
So my question is this:
Is there a notation for this already, like perhaps a $subset$ with a dot in to mean "open subset"
notation
asked Apr 6 '15 at 0:15
Alec TealAlec Teal
3,53812245
$endgroup$
|
show 3 more comments
1
$begingroup$
I very often have to write something like:
$exists U,Vsubseteq M$ where $U,V$ are open, but there's no short hand for it. On my written notes, I do tend to write something like:
$exists U,Vmathop{subseteq}_text{open}M$ is there a common short hand though?
Example of a short hand that exists
$newcommand{bigudot}{mathop{bigcupmkern-14mucdotmkern5mu}}$
$newcommand{udot}{cupmkern-11.5mucdotmkern5mu}$
It is established already that $bigudot$ means "union of sets that are pairwise disjoint", so if I write:
$forall Aexistsmathcal{A}:Asubseteqbigudotmathcal{A}$ - or something - it is clear from the context that $mathcal{A}$ is a family of pairwise disjoint sets
So my question is this:
Is there a notation for this already, like perhaps a $subset$ with a dot in to mean "open subset"
notation
asked Apr 6 '15 at 0:15
Alec TealAlec Teal
3,53812245
$endgroup$
1
$begingroup$
Open is a relative notion, it is not a property of the set. I guess that is why the notation (that I have seen) uses what gives it that property. $Uintau$ where $tau$ is the topology.
$endgroup$
– Karanko
Apr 6 '15 at 0:21
$begingroup$
I am not sure how standard it is, but I have seen it written $U^{text{open}}subset M$.
$endgroup$
– N. Owad
Apr 6 '15 at 0:21
$begingroup$
@Karanko if $M$ above was a topological space, the notion of open becomes clear. What you're saying is "well not even writing 'open' is sufficient, because open is a relative property" - which is absurd
$endgroup$
– Alec Teal
Apr 6 '15 at 0:22
$begingroup$
Absurd is your syllogism that my comment implies that I am saying that "is open" is not sufficient.
$endgroup$
– Karanko
Apr 6 '15 at 0:29
2
$begingroup$
I suspect that whatever your doing in your notes is fine, but elsewhere, there's not all that common a notation - and in most elementary contexts "Let $U$ and $V$ be open subsets in $M$" would be much clearer than any symbol you could introduce (but in other contexts, references to the topology as a pair $(S,tau)$ might be preferred).
$endgroup$
– Milo Brandt
Apr 6 '15 at 0:32
|
show 3 more comments
1
1
1
$begingroup$
I very often have to write something like:
$exists U,Vsubseteq M$ where $U,V$ are open, but there's no short hand for it. On my written notes, I do tend to write something like:
$exists U,Vmathop{subseteq}_text{open}M$ is there a common short hand though?
Example of a short hand that exists
$newcommand{bigudot}{mathop{bigcupmkern-14mucdotmkern5mu}}$
$newcommand{udot}{cupmkern-11.5mucdotmkern5mu}$
It is established already that $bigudot$ means "union of sets that are pairwise disjoint", so if I write:
$forall Aexistsmathcal{A}:Asubseteqbigudotmathcal{A}$ - or something - it is clear from the context that $mathcal{A}$ is a family of pairwise disjoint sets
So my question is this:
Is there a notation for this already, like perhaps a $subset$ with a dot in to mean "open subset"
notation
asked Apr 6 '15 at 0:15
Alec TealAlec Teal
3,53812245
$endgroup$
I very often have to write something like:
$exists U,Vsubseteq M$ where $U,V$ are open, but there's no short hand for it. On my written notes, I do tend to write something like:
$exists U,Vmathop{subseteq}_text{open}M$ is there a common short hand though?
Example of a short hand that exists
$newcommand{bigudot}{mathop{bigcupmkern-14mucdotmkern5mu}}$
$newcommand{udot}{cupmkern-11.5mucdotmkern5mu}$
It is established already that $bigudot$ means "union of sets that are pairwise disjoint", so if I write:
$forall Aexistsmathcal{A}:Asubseteqbigudotmathcal{A}$ - or something - it is clear from the context that $mathcal{A}$ is a family of pairwise disjoint sets
So my question is this:
Is there a notation for this already, like perhaps a $subset$ with a dot in to mean "open subset"
notation
notation
asked Apr 6 '15 at 0:15
Alec TealAlec Teal
3,53812245
asked Apr 6 '15 at 0:15
Alec TealAlec Teal
3,53812245
asked Apr 6 '15 at 0:15
Alec TealAlec Teal
3,53812245
asked Apr 6 '15 at 0:15
Alec TealAlec Teal
3,53812245
asked Apr 6 '15 at 0:15
Alec TealAlec Teal
3,53812245
3,53812245
1
$begingroup$
Open is a relative notion, it is not a property of the set. I guess that is why the notation (that I have seen) uses what gives it that property. $Uintau$ where $tau$ is the topology.
$endgroup$
– Karanko
Apr 6 '15 at 0:21
$begingroup$
I am not sure how standard it is, but I have seen it written $U^{text{open}}subset M$.
$endgroup$
– N. Owad
Apr 6 '15 at 0:21
$begingroup$
@Karanko if $M$ above was a topological space, the notion of open becomes clear. What you're saying is "well not even writing 'open' is sufficient, because open is a relative property" - which is absurd
$endgroup$
– Alec Teal
Apr 6 '15 at 0:22
$begingroup$
Absurd is your syllogism that my comment implies that I am saying that "is open" is not sufficient.
$endgroup$
– Karanko
Apr 6 '15 at 0:29
2
$begingroup$
I suspect that whatever your doing in your notes is fine, but elsewhere, there's not all that common a notation - and in most elementary contexts "Let $U$ and $V$ be open subsets in $M$" would be much clearer than any symbol you could introduce (but in other contexts, references to the topology as a pair $(S,tau)$ might be preferred).
$endgroup$
– Milo Brandt
Apr 6 '15 at 0:32
|
show 3 more comments
1
$begingroup$
Open is a relative notion, it is not a property of the set. I guess that is why the notation (that I have seen) uses what gives it that property. $Uintau$ where $tau$ is the topology.
$endgroup$
– Karanko
Apr 6 '15 at 0:21
$begingroup$
I am not sure how standard it is, but I have seen it written $U^{text{open}}subset M$.
$endgroup$
– N. Owad
Apr 6 '15 at 0:21
$begingroup$
@Karanko if $M$ above was a topological space, the notion of open becomes clear. What you're saying is "well not even writing 'open' is sufficient, because open is a relative property" - which is absurd
$endgroup$
– Alec Teal
Apr 6 '15 at 0:22
$begingroup$
Absurd is your syllogism that my comment implies that I am saying that "is open" is not sufficient.
$endgroup$
– Karanko
Apr 6 '15 at 0:29
2
$begingroup$
I suspect that whatever your doing in your notes is fine, but elsewhere, there's not all that common a notation - and in most elementary contexts "Let $U$ and $V$ be open subsets in $M$" would be much clearer than any symbol you could introduce (but in other contexts, references to the topology as a pair $(S,tau)$ might be preferred).
$endgroup$
– Milo Brandt
Apr 6 '15 at 0:32
1
1
$begingroup$
Open is a relative notion, it is not a property of the set. I guess that is why the notation (that I have seen) uses what gives it that property. $Uintau$ where $tau$ is the topology.
$endgroup$
– Karanko
Apr 6 '15 at 0:21
$begingroup$
Open is a relative notion, it is not a property of the set. I guess that is why the notation (that I have seen) uses what gives it that property. $Uintau$ where $tau$ is the topology.
$endgroup$
– Karanko
Apr 6 '15 at 0:21
$begingroup$
I am not sure how standard it is, but I have seen it written $U^{text{open}}subset M$.
$endgroup$
– N. Owad
Apr 6 '15 at 0:21
$begingroup$
I am not sure how standard it is, but I have seen it written $U^{text{open}}subset M$.
$endgroup$
– N. Owad
Apr 6 '15 at 0:21
$begingroup$
@Karanko if $M$ above was a topological space, the notion of open becomes clear. What you're saying is "well not even writing 'open' is sufficient, because open is a relative property" - which is absurd
$endgroup$
– Alec Teal
Apr 6 '15 at 0:22
$begingroup$
@Karanko if $M$ above was a topological space, the notion of open becomes clear. What you're saying is "well not even writing 'open' is sufficient, because open is a relative property" - which is absurd
$endgroup$
– Alec Teal
Apr 6 '15 at 0:22
$begingroup$
Absurd is your syllogism that my comment implies that I am saying that "is open" is not sufficient.
$endgroup$
– Karanko
Apr 6 '15 at 0:29
$begingroup$
Absurd is your syllogism that my comment implies that I am saying that "is open" is not sufficient.
$endgroup$
– Karanko
Apr 6 '15 at 0:29
2
2
$begingroup$
I suspect that whatever your doing in your notes is fine, but elsewhere, there's not all that common a notation - and in most elementary contexts "Let $U$ and $V$ be open subsets in $M$" would be much clearer than any symbol you could introduce (but in other contexts, references to the topology as a pair $(S,tau)$ might be preferred).
$endgroup$
– Milo Brandt
Apr 6 '15 at 0:32
$begingroup$
I suspect that whatever your doing in your notes is fine, but elsewhere, there's not all that common a notation - and in most elementary contexts "Let $U$ and $V$ be open subsets in $M$" would be much clearer than any symbol you could introduce (but in other contexts, references to the topology as a pair $(S,tau)$ might be preferred).
$endgroup$
– Milo Brandt
Apr 6 '15 at 0:32
|
show 3 more comments
1 Answer
1
active
oldest
votes
0
$begingroup$
Remember that a topological space $X$ is really a pair $(X,mathcal{T})$ where $mathcal{T}$ is a collection of "open" subsets of $X$. We often drop this cumbersome notation. But if you write, $Uin mathcal{T}$, then it is clear you are talking about open sets.
For example, you can define continuity $f:Xto Y$ by saying,
$$ forall Uin mathcal{T}_Y, ~ f^{-1}(U) in mathcal{T}_X $$
answered Apr 6 '15 at 0:21
Nicolas BourbakiNicolas Bourbaki
3,256720
$endgroup$
$begingroup$
I'm asking for a subset notation. Which topology? The subset topology or open WRT the parent topology? Also writing $Usubset Vwedge Uinmathcal{J}$ isn't really a short hand symbol
$endgroup$
– Alec Teal
Apr 6 '15 at 0:28
$begingroup$
Furthermore! We rarely explicitly mention the topology - even when studying topology the $(X,mathcal{J})$ notation was quickly dropped. IME anyway
$endgroup$
– Alec Teal
Apr 6 '15 at 0:30
$begingroup$
If $A$ is an open subset of $B$, where $B$ is a subset of $X$ (topological space) then you can simply write $Ain mathcal{T}_B$, and it becomes clear that $mathcal{T}_B$ is referring to the topology on $B$ and not on $X$.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:35
$begingroup$
See that's missing the point, and it's not really a "convention", I would use $mathcal{J}_X$ for a topology on $X$ say, you would use T - see the cup with a dot in it example which does away with the words "pairwise disjoint" - I'm also reluctant to mention a specific topology because it doesn't matter, for the same reason the neighbourhood part is dropped when defining germs - the neighbourhood doesn't matter.
$endgroup$
– Alec Teal
Apr 6 '15 at 0:38
$begingroup$
I never seen your dot-union notation before to mean disjoint union. As far as I know, there is no universal notation for disjoint union. Most people use a square-shaped U but every author simply just explains that it is a disjoint union. The same is true with "open". There is no universal notation for open, whenever topological arguments become delicate authors just spell out which set is open where. I think you are over-thinking this notation, in the end it does not matter so much.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:41
|
show 1 more comment
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
0
$begingroup$
Remember that a topological space $X$ is really a pair $(X,mathcal{T})$ where $mathcal{T}$ is a collection of "open" subsets of $X$. We often drop this cumbersome notation. But if you write, $Uin mathcal{T}$, then it is clear you are talking about open sets.
For example, you can define continuity $f:Xto Y$ by saying,
$$ forall Uin mathcal{T}_Y, ~ f^{-1}(U) in mathcal{T}_X $$
answered Apr 6 '15 at 0:21
Nicolas BourbakiNicolas Bourbaki
3,256720
$endgroup$
$begingroup$
I'm asking for a subset notation. Which topology? The subset topology or open WRT the parent topology? Also writing $Usubset Vwedge Uinmathcal{J}$ isn't really a short hand symbol
$endgroup$
– Alec Teal
Apr 6 '15 at 0:28
$begingroup$
Furthermore! We rarely explicitly mention the topology - even when studying topology the $(X,mathcal{J})$ notation was quickly dropped. IME anyway
$endgroup$
– Alec Teal
Apr 6 '15 at 0:30
$begingroup$
If $A$ is an open subset of $B$, where $B$ is a subset of $X$ (topological space) then you can simply write $Ain mathcal{T}_B$, and it becomes clear that $mathcal{T}_B$ is referring to the topology on $B$ and not on $X$.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:35
$begingroup$
See that's missing the point, and it's not really a "convention", I would use $mathcal{J}_X$ for a topology on $X$ say, you would use T - see the cup with a dot in it example which does away with the words "pairwise disjoint" - I'm also reluctant to mention a specific topology because it doesn't matter, for the same reason the neighbourhood part is dropped when defining germs - the neighbourhood doesn't matter.
$endgroup$
– Alec Teal
Apr 6 '15 at 0:38
$begingroup$
I never seen your dot-union notation before to mean disjoint union. As far as I know, there is no universal notation for disjoint union. Most people use a square-shaped U but every author simply just explains that it is a disjoint union. The same is true with "open". There is no universal notation for open, whenever topological arguments become delicate authors just spell out which set is open where. I think you are over-thinking this notation, in the end it does not matter so much.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:41
|
show 1 more comment
0
$begingroup$
Remember that a topological space $X$ is really a pair $(X,mathcal{T})$ where $mathcal{T}$ is a collection of "open" subsets of $X$. We often drop this cumbersome notation. But if you write, $Uin mathcal{T}$, then it is clear you are talking about open sets.
For example, you can define continuity $f:Xto Y$ by saying,
$$ forall Uin mathcal{T}_Y, ~ f^{-1}(U) in mathcal{T}_X $$
answered Apr 6 '15 at 0:21
Nicolas BourbakiNicolas Bourbaki
3,256720
$endgroup$
$begingroup$
I'm asking for a subset notation. Which topology? The subset topology or open WRT the parent topology? Also writing $Usubset Vwedge Uinmathcal{J}$ isn't really a short hand symbol
$endgroup$
– Alec Teal
Apr 6 '15 at 0:28
$begingroup$
Furthermore! We rarely explicitly mention the topology - even when studying topology the $(X,mathcal{J})$ notation was quickly dropped. IME anyway
$endgroup$
– Alec Teal
Apr 6 '15 at 0:30
$begingroup$
If $A$ is an open subset of $B$, where $B$ is a subset of $X$ (topological space) then you can simply write $Ain mathcal{T}_B$, and it becomes clear that $mathcal{T}_B$ is referring to the topology on $B$ and not on $X$.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:35
$begingroup$
See that's missing the point, and it's not really a "convention", I would use $mathcal{J}_X$ for a topology on $X$ say, you would use T - see the cup with a dot in it example which does away with the words "pairwise disjoint" - I'm also reluctant to mention a specific topology because it doesn't matter, for the same reason the neighbourhood part is dropped when defining germs - the neighbourhood doesn't matter.
$endgroup$
– Alec Teal
Apr 6 '15 at 0:38
$begingroup$
I never seen your dot-union notation before to mean disjoint union. As far as I know, there is no universal notation for disjoint union. Most people use a square-shaped U but every author simply just explains that it is a disjoint union. The same is true with "open". There is no universal notation for open, whenever topological arguments become delicate authors just spell out which set is open where. I think you are over-thinking this notation, in the end it does not matter so much.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:41
|
show 1 more comment
0
0
0
$begingroup$
Remember that a topological space $X$ is really a pair $(X,mathcal{T})$ where $mathcal{T}$ is a collection of "open" subsets of $X$. We often drop this cumbersome notation. But if you write, $Uin mathcal{T}$, then it is clear you are talking about open sets.
For example, you can define continuity $f:Xto Y$ by saying,
$$ forall Uin mathcal{T}_Y, ~ f^{-1}(U) in mathcal{T}_X $$
answered Apr 6 '15 at 0:21
Nicolas BourbakiNicolas Bourbaki
3,256720
$endgroup$
Remember that a topological space $X$ is really a pair $(X,mathcal{T})$ where $mathcal{T}$ is a collection of "open" subsets of $X$. We often drop this cumbersome notation. But if you write, $Uin mathcal{T}$, then it is clear you are talking about open sets.
For example, you can define continuity $f:Xto Y$ by saying,
$$ forall Uin mathcal{T}_Y, ~ f^{-1}(U) in mathcal{T}_X $$
answered Apr 6 '15 at 0:21
Nicolas BourbakiNicolas Bourbaki
3,256720
answered Apr 6 '15 at 0:21
Nicolas BourbakiNicolas Bourbaki
3,256720
answered Apr 6 '15 at 0:21
Nicolas BourbakiNicolas Bourbaki
3,256720
answered Apr 6 '15 at 0:21
Nicolas BourbakiNicolas Bourbaki
3,256720
3,256720
$begingroup$
I'm asking for a subset notation. Which topology? The subset topology or open WRT the parent topology? Also writing $Usubset Vwedge Uinmathcal{J}$ isn't really a short hand symbol
$endgroup$
– Alec Teal
Apr 6 '15 at 0:28
$begingroup$
Furthermore! We rarely explicitly mention the topology - even when studying topology the $(X,mathcal{J})$ notation was quickly dropped. IME anyway
$endgroup$
– Alec Teal
Apr 6 '15 at 0:30
$begingroup$
If $A$ is an open subset of $B$, where $B$ is a subset of $X$ (topological space) then you can simply write $Ain mathcal{T}_B$, and it becomes clear that $mathcal{T}_B$ is referring to the topology on $B$ and not on $X$.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:35
$begingroup$
See that's missing the point, and it's not really a "convention", I would use $mathcal{J}_X$ for a topology on $X$ say, you would use T - see the cup with a dot in it example which does away with the words "pairwise disjoint" - I'm also reluctant to mention a specific topology because it doesn't matter, for the same reason the neighbourhood part is dropped when defining germs - the neighbourhood doesn't matter.
$endgroup$
– Alec Teal
Apr 6 '15 at 0:38
$begingroup$
I never seen your dot-union notation before to mean disjoint union. As far as I know, there is no universal notation for disjoint union. Most people use a square-shaped U but every author simply just explains that it is a disjoint union. The same is true with "open". There is no universal notation for open, whenever topological arguments become delicate authors just spell out which set is open where. I think you are over-thinking this notation, in the end it does not matter so much.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:41
|
show 1 more comment
$begingroup$
I'm asking for a subset notation. Which topology? The subset topology or open WRT the parent topology? Also writing $Usubset Vwedge Uinmathcal{J}$ isn't really a short hand symbol
$endgroup$
– Alec Teal
Apr 6 '15 at 0:28
$begingroup$
Furthermore! We rarely explicitly mention the topology - even when studying topology the $(X,mathcal{J})$ notation was quickly dropped. IME anyway
$endgroup$
– Alec Teal
Apr 6 '15 at 0:30
$begingroup$
If $A$ is an open subset of $B$, where $B$ is a subset of $X$ (topological space) then you can simply write $Ain mathcal{T}_B$, and it becomes clear that $mathcal{T}_B$ is referring to the topology on $B$ and not on $X$.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:35
$begingroup$
See that's missing the point, and it's not really a "convention", I would use $mathcal{J}_X$ for a topology on $X$ say, you would use T - see the cup with a dot in it example which does away with the words "pairwise disjoint" - I'm also reluctant to mention a specific topology because it doesn't matter, for the same reason the neighbourhood part is dropped when defining germs - the neighbourhood doesn't matter.
$endgroup$
– Alec Teal
Apr 6 '15 at 0:38
$begingroup$
I never seen your dot-union notation before to mean disjoint union. As far as I know, there is no universal notation for disjoint union. Most people use a square-shaped U but every author simply just explains that it is a disjoint union. The same is true with "open". There is no universal notation for open, whenever topological arguments become delicate authors just spell out which set is open where. I think you are over-thinking this notation, in the end it does not matter so much.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:41
$begingroup$
I'm asking for a subset notation. Which topology? The subset topology or open WRT the parent topology? Also writing $Usubset Vwedge Uinmathcal{J}$ isn't really a short hand symbol
$endgroup$
– Alec Teal
Apr 6 '15 at 0:28
$begingroup$
I'm asking for a subset notation. Which topology? The subset topology or open WRT the parent topology? Also writing $Usubset Vwedge Uinmathcal{J}$ isn't really a short hand symbol
$endgroup$
– Alec Teal
Apr 6 '15 at 0:28
$begingroup$
Furthermore! We rarely explicitly mention the topology - even when studying topology the $(X,mathcal{J})$ notation was quickly dropped. IME anyway
$endgroup$
– Alec Teal
Apr 6 '15 at 0:30
$begingroup$
Furthermore! We rarely explicitly mention the topology - even when studying topology the $(X,mathcal{J})$ notation was quickly dropped. IME anyway
$endgroup$
– Alec Teal
Apr 6 '15 at 0:30
$begingroup$
If $A$ is an open subset of $B$, where $B$ is a subset of $X$ (topological space) then you can simply write $Ain mathcal{T}_B$, and it becomes clear that $mathcal{T}_B$ is referring to the topology on $B$ and not on $X$.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:35
$begingroup$
If $A$ is an open subset of $B$, where $B$ is a subset of $X$ (topological space) then you can simply write $Ain mathcal{T}_B$, and it becomes clear that $mathcal{T}_B$ is referring to the topology on $B$ and not on $X$.
$endgroup$
– Nicolas Bourbaki
Apr 6 '15 at 0:35
$begingroup$
See that's missing the point, and it's not really a "convention", I would use $mathcal{J}_X$ for a topology on $X$ say, you would use T - see the cup with a dot in it example which does away with the words "pairwise disjoint" - I'm also reluctant to mention a specific topology because it doesn't matter, for the same reason the neighbourhood part is dropped when defining germs - the neighbourhood doesn't matter.
$endgroup$
– Alec Teal
Apr 6 '15 at 0:38
$begingroup$
See that's missing the point, and it's not really a "convention", I would use $mathcal{J}_X$ for a topology on $X$ say, you would use T - see the cup with a dot in it example which does away with the words "pairwise disjoint" - I'm also reluctant to mention a specific topology because it doesn't matter, for the same reason the neighbourhood part is dropped when defining germs - the neighbourhood doesn't matter.
$endgroup$
– Alec Teal
Apr 6 '15 at 0:38
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I never seen your dot-union notation before to mean disjoint union. As far as I know, there is no universal notation for disjoint union. Most people use a square-shaped U but every author simply just explains that it is a disjoint union. The same is true with "open". There is no universal notation for open, whenever topological arguments become delicate authors just spell out which set is open where. I think you are over-thinking this notation, in the end it does not matter so much.
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– Nicolas Bourbaki
Apr 6 '15 at 0:41
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I never seen your dot-union notation before to mean disjoint union. As far as I know, there is no universal notation for disjoint union. Most people use a square-shaped U but every author simply just explains that it is a disjoint union. The same is true with "open". There is no universal notation for open, whenever topological arguments become delicate authors just spell out which set is open where. I think you are over-thinking this notation, in the end it does not matter so much.
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– Nicolas Bourbaki
Apr 6 '15 at 0:41
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1
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Open is a relative notion, it is not a property of the set. I guess that is why the notation (that I have seen) uses what gives it that property. $Uintau$ where $tau$ is the topology.
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– Karanko
Apr 6 '15 at 0:21
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I am not sure how standard it is, but I have seen it written $U^{text{open}}subset M$.
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– N. Owad
Apr 6 '15 at 0:21
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@Karanko if $M$ above was a topological space, the notion of open becomes clear. What you're saying is "well not even writing 'open' is sufficient, because open is a relative property" - which is absurd
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– Alec Teal
Apr 6 '15 at 0:22
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Absurd is your syllogism that my comment implies that I am saying that "is open" is not sufficient.
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– Karanko
Apr 6 '15 at 0:29
2
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I suspect that whatever your doing in your notes is fine, but elsewhere, there's not all that common a notation - and in most elementary contexts "Let $U$ and $V$ be open subsets in $M$" would be much clearer than any symbol you could introduce (but in other contexts, references to the topology as a pair $(S,tau)$ might be preferred).
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– Milo Brandt
Apr 6 '15 at 0:32