What are some interesting applications or properties of the topological residue of a set? Is it too similar to the boundary of a set?Properties of a set (closure, interior, boundary)$X$ a topological space. If $A$ lies inside a closed set. Does it follow that the closure of $A$ also lies inside this closed set?Some topological properties of bounded domainsOn the “regularity” of the boundary of an open setWhy is the boundary of a set in the closure of the set?Name of the set $B:= overlineAsetminus A$What is the boundary of $mathbbQ times mathbbQ$ in $mathbbR times mathbbQ$?Boundary of a Subset is Closed in a SpaceTopology, equivalent statements, nets, boundaryInterior of the boundary of an open set is empty
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What are some interesting applications or properties of the topological residue of a set? Is it too similar to the boundary of a set?
Properties of a set (closure, interior, boundary)$X$ a topological space. If $A$ lies inside a closed set. Does it follow that the closure of $A$ also lies inside this closed set?Some topological properties of bounded domainsOn the “regularity” of the boundary of an open setWhy is the boundary of a set in the closure of the set?Name of the set $B:= overlineAsetminus A$What is the boundary of $mathbbQ times mathbbQ$ in $mathbbR times mathbbQ$?Boundary of a Subset is Closed in a SpaceTopology, equivalent statements, nets, boundaryInterior of the boundary of an open set is empty
$begingroup$
Recently I stumbled upon the concept of residue of a set (alternately known as the frontier of a set) in general topology, which is the closure of a set A without A itself, or in other words all of the boundary points of A not contained in A:
$$operatornameRes(A)= overline A setminus A$$
I'm not quite sure who first came up with this idea, so I'm not able to provide any useful references. This is very similar to the boundary of A, which for context is usually defined as the closure minus the interior:
$$ partial(A)= overline A setminus A^circ$$
I'm trying to find applications for this concept, so I've been looking for some properties of this residue operation, such as how unions and intersections of sets behave under this operation, how many different sets I can obtain using only the union and residue operations, whether I can use this notion as a primitive for defining topological spaces or weaker notions of it, etc.
So far I have the following:
beginalign
operatornameRes(A) subseteq & partial A \
operatornameRes(C)=&varnothing Leftrightarrow textC is closed \
operatornameRes(U)=&partial(U) Leftrightarrow textU is open \
complement_X(operatornameRes(A))&=A cup operatornameExt(A) \
operatornameResleft(operatornameRes(A)right)&cap operatornameRes(A)= varnothing quad *
endalign
*(hence each successive iteration of $operatornameRes$ will be disjoint from the previous one.)
Just to be absolutely clear, here I define the exterior of A as:
$$operatornameExt(A)=complement_Xleft(overline Aright)=left(complement_X(A)right)^circ$$
This is pretty tedious work, and I realize to many this is pretty trivial and not all that interesting, but I just want to see how much I can get out of this residue operator and whether it has any interesting connections to other topological notions.
general-topology
asked Mar 6 at 18:36
Quantum ChillQuantum Chill
718
$endgroup$
add a comment |
$begingroup$
Recently I stumbled upon the concept of residue of a set (alternately known as the frontier of a set) in general topology, which is the closure of a set A without A itself, or in other words all of the boundary points of A not contained in A:
$$operatornameRes(A)= overline A setminus A$$
I'm not quite sure who first came up with this idea, so I'm not able to provide any useful references. This is very similar to the boundary of A, which for context is usually defined as the closure minus the interior:
$$ partial(A)= overline A setminus A^circ$$
I'm trying to find applications for this concept, so I've been looking for some properties of this residue operation, such as how unions and intersections of sets behave under this operation, how many different sets I can obtain using only the union and residue operations, whether I can use this notion as a primitive for defining topological spaces or weaker notions of it, etc.
So far I have the following:
beginalign
operatornameRes(A) subseteq & partial A \
operatornameRes(C)=&varnothing Leftrightarrow textC is closed \
operatornameRes(U)=&partial(U) Leftrightarrow textU is open \
complement_X(operatornameRes(A))&=A cup operatornameExt(A) \
operatornameResleft(operatornameRes(A)right)&cap operatornameRes(A)= varnothing quad *
endalign
*(hence each successive iteration of $operatornameRes$ will be disjoint from the previous one.)
Just to be absolutely clear, here I define the exterior of A as:
$$operatornameExt(A)=complement_Xleft(overline Aright)=left(complement_X(A)right)^circ$$
This is pretty tedious work, and I realize to many this is pretty trivial and not all that interesting, but I just want to see how much I can get out of this residue operator and whether it has any interesting connections to other topological notions.
general-topology
asked Mar 6 at 18:36
Quantum ChillQuantum Chill
718
$endgroup$
$begingroup$
I don't know anything about residues, but one thing you might consider is the following: given a particular set how many different sets can you get to using the residue, complement, interior, and closure operators? I believe with closure, complement, and interior the answer is 14. Maybe you can get all sets if you add residue?
$endgroup$
– Robert Thingum
Mar 7 at 1:07
$begingroup$
@RobertThingum I actually did something similar with the boundary, building sets only using A, the boundary operator, and union. There I got 13 distinct sets (not counting the empty set), so I might try the same with the residue operator. Thanks a bunch for the suggestion!
$endgroup$
– Quantum Chill
Mar 7 at 13:49
$begingroup$
In the event you can get every set (doubtful). You should potentially design a structure with operations that act as residue, interior, etc. that is in all but the literal sense a topological space. The study of frames and locales is somewhat like this.
$endgroup$
– Robert Thingum
Mar 7 at 13:54
1
$begingroup$
A closely related concept is defined in: Ronald Brown, Topology and Groupoids, Ex. 2.2.8. What is usually called the boundary of a subset is here called its frontier, and what you call the residue of a subset is what here would be called the boundary of its complement: "For any subset $A$ of a topological space $X$, define [...] $operatornameBdA$ (the boundary of $A$) [...] as follows: [...] $operatornameBdA=A setminusoperatornameIntA$ [...]" There follows a list of identities to be proved, including some involving $operatornameBdA$.
$endgroup$
– Calum Gilhooley
Mar 11 at 22:22
$begingroup$
@CalumGilhooley Thanks for the reference! I actually just defined the above concept of boundary myself, so far I'm calling it the perimeter of $A$, and I found that I can represent my notion of boundary of $A$ as a disjoint union of the residue and perimeter. These two sets seem to be dual to each other in a way, as the perimeter is empty for open sets and the residue is empty for closed sets.
$endgroup$
– Quantum Chill
Mar 12 at 14:49
add a comment |
$begingroup$
Recently I stumbled upon the concept of residue of a set (alternately known as the frontier of a set) in general topology, which is the closure of a set A without A itself, or in other words all of the boundary points of A not contained in A:
$$operatornameRes(A)= overline A setminus A$$
I'm not quite sure who first came up with this idea, so I'm not able to provide any useful references. This is very similar to the boundary of A, which for context is usually defined as the closure minus the interior:
$$ partial(A)= overline A setminus A^circ$$
I'm trying to find applications for this concept, so I've been looking for some properties of this residue operation, such as how unions and intersections of sets behave under this operation, how many different sets I can obtain using only the union and residue operations, whether I can use this notion as a primitive for defining topological spaces or weaker notions of it, etc.
So far I have the following:
beginalign
operatornameRes(A) subseteq & partial A \
operatornameRes(C)=&varnothing Leftrightarrow textC is closed \
operatornameRes(U)=&partial(U) Leftrightarrow textU is open \
complement_X(operatornameRes(A))&=A cup operatornameExt(A) \
operatornameResleft(operatornameRes(A)right)&cap operatornameRes(A)= varnothing quad *
endalign
*(hence each successive iteration of $operatornameRes$ will be disjoint from the previous one.)
Just to be absolutely clear, here I define the exterior of A as:
$$operatornameExt(A)=complement_Xleft(overline Aright)=left(complement_X(A)right)^circ$$
This is pretty tedious work, and I realize to many this is pretty trivial and not all that interesting, but I just want to see how much I can get out of this residue operator and whether it has any interesting connections to other topological notions.
general-topology
asked Mar 6 at 18:36
Quantum ChillQuantum Chill
718
$endgroup$
Recently I stumbled upon the concept of residue of a set (alternately known as the frontier of a set) in general topology, which is the closure of a set A without A itself, or in other words all of the boundary points of A not contained in A:
$$operatornameRes(A)= overline A setminus A$$
I'm not quite sure who first came up with this idea, so I'm not able to provide any useful references. This is very similar to the boundary of A, which for context is usually defined as the closure minus the interior:
$$ partial(A)= overline A setminus A^circ$$
I'm trying to find applications for this concept, so I've been looking for some properties of this residue operation, such as how unions and intersections of sets behave under this operation, how many different sets I can obtain using only the union and residue operations, whether I can use this notion as a primitive for defining topological spaces or weaker notions of it, etc.
So far I have the following:
beginalign
operatornameRes(A) subseteq & partial A \
operatornameRes(C)=&varnothing Leftrightarrow textC is closed \
operatornameRes(U)=&partial(U) Leftrightarrow textU is open \
complement_X(operatornameRes(A))&=A cup operatornameExt(A) \
operatornameResleft(operatornameRes(A)right)&cap operatornameRes(A)= varnothing quad *
endalign
*(hence each successive iteration of $operatornameRes$ will be disjoint from the previous one.)
Just to be absolutely clear, here I define the exterior of A as:
$$operatornameExt(A)=complement_Xleft(overline Aright)=left(complement_X(A)right)^circ$$
This is pretty tedious work, and I realize to many this is pretty trivial and not all that interesting, but I just want to see how much I can get out of this residue operator and whether it has any interesting connections to other topological notions.
general-topology
general-topology
asked Mar 6 at 18:36
Quantum ChillQuantum Chill
718
asked Mar 6 at 18:36
Quantum ChillQuantum Chill
718
edited Mar 11 at 21:08
Quantum Chill
asked Mar 6 at 18:36
Quantum ChillQuantum Chill
718
asked Mar 6 at 18:36
Quantum ChillQuantum Chill
718
asked Mar 6 at 18:36
Quantum ChillQuantum Chill
718
718
$begingroup$
I don't know anything about residues, but one thing you might consider is the following: given a particular set how many different sets can you get to using the residue, complement, interior, and closure operators? I believe with closure, complement, and interior the answer is 14. Maybe you can get all sets if you add residue?
$endgroup$
– Robert Thingum
Mar 7 at 1:07
$begingroup$
@RobertThingum I actually did something similar with the boundary, building sets only using A, the boundary operator, and union. There I got 13 distinct sets (not counting the empty set), so I might try the same with the residue operator. Thanks a bunch for the suggestion!
$endgroup$
– Quantum Chill
Mar 7 at 13:49
$begingroup$
In the event you can get every set (doubtful). You should potentially design a structure with operations that act as residue, interior, etc. that is in all but the literal sense a topological space. The study of frames and locales is somewhat like this.
$endgroup$
– Robert Thingum
Mar 7 at 13:54
1
$begingroup$
A closely related concept is defined in: Ronald Brown, Topology and Groupoids, Ex. 2.2.8. What is usually called the boundary of a subset is here called its frontier, and what you call the residue of a subset is what here would be called the boundary of its complement: "For any subset $A$ of a topological space $X$, define [...] $operatornameBdA$ (the boundary of $A$) [...] as follows: [...] $operatornameBdA=A setminusoperatornameIntA$ [...]" There follows a list of identities to be proved, including some involving $operatornameBdA$.
$endgroup$
– Calum Gilhooley
Mar 11 at 22:22
$begingroup$
@CalumGilhooley Thanks for the reference! I actually just defined the above concept of boundary myself, so far I'm calling it the perimeter of $A$, and I found that I can represent my notion of boundary of $A$ as a disjoint union of the residue and perimeter. These two sets seem to be dual to each other in a way, as the perimeter is empty for open sets and the residue is empty for closed sets.
$endgroup$
– Quantum Chill
Mar 12 at 14:49
add a comment |
$begingroup$
I don't know anything about residues, but one thing you might consider is the following: given a particular set how many different sets can you get to using the residue, complement, interior, and closure operators? I believe with closure, complement, and interior the answer is 14. Maybe you can get all sets if you add residue?
$endgroup$
– Robert Thingum
Mar 7 at 1:07
$begingroup$
@RobertThingum I actually did something similar with the boundary, building sets only using A, the boundary operator, and union. There I got 13 distinct sets (not counting the empty set), so I might try the same with the residue operator. Thanks a bunch for the suggestion!
$endgroup$
– Quantum Chill
Mar 7 at 13:49
$begingroup$
In the event you can get every set (doubtful). You should potentially design a structure with operations that act as residue, interior, etc. that is in all but the literal sense a topological space. The study of frames and locales is somewhat like this.
$endgroup$
– Robert Thingum
Mar 7 at 13:54
1
$begingroup$
A closely related concept is defined in: Ronald Brown, Topology and Groupoids, Ex. 2.2.8. What is usually called the boundary of a subset is here called its frontier, and what you call the residue of a subset is what here would be called the boundary of its complement: "For any subset $A$ of a topological space $X$, define [...] $operatornameBdA$ (the boundary of $A$) [...] as follows: [...] $operatornameBdA=A setminusoperatornameIntA$ [...]" There follows a list of identities to be proved, including some involving $operatornameBdA$.
$endgroup$
– Calum Gilhooley
Mar 11 at 22:22
$begingroup$
@CalumGilhooley Thanks for the reference! I actually just defined the above concept of boundary myself, so far I'm calling it the perimeter of $A$, and I found that I can represent my notion of boundary of $A$ as a disjoint union of the residue and perimeter. These two sets seem to be dual to each other in a way, as the perimeter is empty for open sets and the residue is empty for closed sets.
$endgroup$
– Quantum Chill
Mar 12 at 14:49
$begingroup$
I don't know anything about residues, but one thing you might consider is the following: given a particular set how many different sets can you get to using the residue, complement, interior, and closure operators? I believe with closure, complement, and interior the answer is 14. Maybe you can get all sets if you add residue?
$endgroup$
– Robert Thingum
Mar 7 at 1:07
$begingroup$
I don't know anything about residues, but one thing you might consider is the following: given a particular set how many different sets can you get to using the residue, complement, interior, and closure operators? I believe with closure, complement, and interior the answer is 14. Maybe you can get all sets if you add residue?
$endgroup$
– Robert Thingum
Mar 7 at 1:07
$begingroup$
@RobertThingum I actually did something similar with the boundary, building sets only using A, the boundary operator, and union. There I got 13 distinct sets (not counting the empty set), so I might try the same with the residue operator. Thanks a bunch for the suggestion!
$endgroup$
– Quantum Chill
Mar 7 at 13:49
$begingroup$
@RobertThingum I actually did something similar with the boundary, building sets only using A, the boundary operator, and union. There I got 13 distinct sets (not counting the empty set), so I might try the same with the residue operator. Thanks a bunch for the suggestion!
$endgroup$
– Quantum Chill
Mar 7 at 13:49
$begingroup$
In the event you can get every set (doubtful). You should potentially design a structure with operations that act as residue, interior, etc. that is in all but the literal sense a topological space. The study of frames and locales is somewhat like this.
$endgroup$
– Robert Thingum
Mar 7 at 13:54
$begingroup$
In the event you can get every set (doubtful). You should potentially design a structure with operations that act as residue, interior, etc. that is in all but the literal sense a topological space. The study of frames and locales is somewhat like this.
$endgroup$
– Robert Thingum
Mar 7 at 13:54
1
1
$begingroup$
A closely related concept is defined in: Ronald Brown, Topology and Groupoids, Ex. 2.2.8. What is usually called the boundary of a subset is here called its frontier, and what you call the residue of a subset is what here would be called the boundary of its complement: "For any subset $A$ of a topological space $X$, define [...] $operatornameBdA$ (the boundary of $A$) [...] as follows: [...] $operatornameBdA=A setminusoperatornameIntA$ [...]" There follows a list of identities to be proved, including some involving $operatornameBdA$.
$endgroup$
– Calum Gilhooley
Mar 11 at 22:22
$begingroup$
A closely related concept is defined in: Ronald Brown, Topology and Groupoids, Ex. 2.2.8. What is usually called the boundary of a subset is here called its frontier, and what you call the residue of a subset is what here would be called the boundary of its complement: "For any subset $A$ of a topological space $X$, define [...] $operatornameBdA$ (the boundary of $A$) [...] as follows: [...] $operatornameBdA=A setminusoperatornameIntA$ [...]" There follows a list of identities to be proved, including some involving $operatornameBdA$.
$endgroup$
– Calum Gilhooley
Mar 11 at 22:22
$begingroup$
@CalumGilhooley Thanks for the reference! I actually just defined the above concept of boundary myself, so far I'm calling it the perimeter of $A$, and I found that I can represent my notion of boundary of $A$ as a disjoint union of the residue and perimeter. These two sets seem to be dual to each other in a way, as the perimeter is empty for open sets and the residue is empty for closed sets.
$endgroup$
– Quantum Chill
Mar 12 at 14:49
$begingroup$
@CalumGilhooley Thanks for the reference! I actually just defined the above concept of boundary myself, so far I'm calling it the perimeter of $A$, and I found that I can represent my notion of boundary of $A$ as a disjoint union of the residue and perimeter. These two sets seem to be dual to each other in a way, as the perimeter is empty for open sets and the residue is empty for closed sets.
$endgroup$
– Quantum Chill
Mar 12 at 14:49
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
More of a response to the comments than a direct answer to the question, although the comments suggested this could count as an application:
You can get arbitrarily many sets by iterating the residue operator.
If $X_1$ is the unit circle minus a point then $operatornameRes X_1$ is a point.
If $X_2=Dsetminus X_1$, where D is the unit disc, then $operatornameRes X_2 = X_1$. So iterating the residue operator gives you four sets ($X_2$,$X_1$, point, empty set).
You can build more sets $X_n$ in an analogous manner, getting arbitrarily large numbers of sets you can reach by iterated residues alone.
These sorts of things come up with lexicographic orders, so you might want to see if there's some useful connection.
answered Mar 11 at 21:54
Daniel McLauryDaniel McLaury
15.9k32981
$endgroup$
$begingroup$
I had the suspicion that this would be the case, since the sequence of iterations would go on until we reach a closed set, if we ever reach it. Maybe there exists a concrete class of spaces where iterating the residue of an arbitrary subset results in finitely many sets? Now I gotta brush up on the order topology, cheers!
$endgroup$
– Quantum Chill
Mar 12 at 14:55
add a comment |
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$begingroup$
More of a response to the comments than a direct answer to the question, although the comments suggested this could count as an application:
You can get arbitrarily many sets by iterating the residue operator.
If $X_1$ is the unit circle minus a point then $operatornameRes X_1$ is a point.
If $X_2=Dsetminus X_1$, where D is the unit disc, then $operatornameRes X_2 = X_1$. So iterating the residue operator gives you four sets ($X_2$,$X_1$, point, empty set).
You can build more sets $X_n$ in an analogous manner, getting arbitrarily large numbers of sets you can reach by iterated residues alone.
These sorts of things come up with lexicographic orders, so you might want to see if there's some useful connection.
answered Mar 11 at 21:54
Daniel McLauryDaniel McLaury
15.9k32981
$endgroup$
$begingroup$
I had the suspicion that this would be the case, since the sequence of iterations would go on until we reach a closed set, if we ever reach it. Maybe there exists a concrete class of spaces where iterating the residue of an arbitrary subset results in finitely many sets? Now I gotta brush up on the order topology, cheers!
$endgroup$
– Quantum Chill
Mar 12 at 14:55
add a comment |
$begingroup$
More of a response to the comments than a direct answer to the question, although the comments suggested this could count as an application:
You can get arbitrarily many sets by iterating the residue operator.
If $X_1$ is the unit circle minus a point then $operatornameRes X_1$ is a point.
If $X_2=Dsetminus X_1$, where D is the unit disc, then $operatornameRes X_2 = X_1$. So iterating the residue operator gives you four sets ($X_2$,$X_1$, point, empty set).
You can build more sets $X_n$ in an analogous manner, getting arbitrarily large numbers of sets you can reach by iterated residues alone.
These sorts of things come up with lexicographic orders, so you might want to see if there's some useful connection.
answered Mar 11 at 21:54
Daniel McLauryDaniel McLaury
15.9k32981
$endgroup$
$begingroup$
I had the suspicion that this would be the case, since the sequence of iterations would go on until we reach a closed set, if we ever reach it. Maybe there exists a concrete class of spaces where iterating the residue of an arbitrary subset results in finitely many sets? Now I gotta brush up on the order topology, cheers!
$endgroup$
– Quantum Chill
Mar 12 at 14:55
add a comment |
$begingroup$
More of a response to the comments than a direct answer to the question, although the comments suggested this could count as an application:
You can get arbitrarily many sets by iterating the residue operator.
If $X_1$ is the unit circle minus a point then $operatornameRes X_1$ is a point.
If $X_2=Dsetminus X_1$, where D is the unit disc, then $operatornameRes X_2 = X_1$. So iterating the residue operator gives you four sets ($X_2$,$X_1$, point, empty set).
You can build more sets $X_n$ in an analogous manner, getting arbitrarily large numbers of sets you can reach by iterated residues alone.
These sorts of things come up with lexicographic orders, so you might want to see if there's some useful connection.
answered Mar 11 at 21:54
Daniel McLauryDaniel McLaury
15.9k32981
$endgroup$
More of a response to the comments than a direct answer to the question, although the comments suggested this could count as an application:
You can get arbitrarily many sets by iterating the residue operator.
If $X_1$ is the unit circle minus a point then $operatornameRes X_1$ is a point.
If $X_2=Dsetminus X_1$, where D is the unit disc, then $operatornameRes X_2 = X_1$. So iterating the residue operator gives you four sets ($X_2$,$X_1$, point, empty set).
You can build more sets $X_n$ in an analogous manner, getting arbitrarily large numbers of sets you can reach by iterated residues alone.
These sorts of things come up with lexicographic orders, so you might want to see if there's some useful connection.
answered Mar 11 at 21:54
Daniel McLauryDaniel McLaury
15.9k32981
answered Mar 11 at 21:54
Daniel McLauryDaniel McLaury
15.9k32981
answered Mar 11 at 21:54
Daniel McLauryDaniel McLaury
15.9k32981
answered Mar 11 at 21:54
Daniel McLauryDaniel McLaury
15.9k32981
15.9k32981
$begingroup$
I had the suspicion that this would be the case, since the sequence of iterations would go on until we reach a closed set, if we ever reach it. Maybe there exists a concrete class of spaces where iterating the residue of an arbitrary subset results in finitely many sets? Now I gotta brush up on the order topology, cheers!
$endgroup$
– Quantum Chill
Mar 12 at 14:55
add a comment |
$begingroup$
I had the suspicion that this would be the case, since the sequence of iterations would go on until we reach a closed set, if we ever reach it. Maybe there exists a concrete class of spaces where iterating the residue of an arbitrary subset results in finitely many sets? Now I gotta brush up on the order topology, cheers!
$endgroup$
– Quantum Chill
Mar 12 at 14:55
$begingroup$
I had the suspicion that this would be the case, since the sequence of iterations would go on until we reach a closed set, if we ever reach it. Maybe there exists a concrete class of spaces where iterating the residue of an arbitrary subset results in finitely many sets? Now I gotta brush up on the order topology, cheers!
$endgroup$
– Quantum Chill
Mar 12 at 14:55
$begingroup$
I had the suspicion that this would be the case, since the sequence of iterations would go on until we reach a closed set, if we ever reach it. Maybe there exists a concrete class of spaces where iterating the residue of an arbitrary subset results in finitely many sets? Now I gotta brush up on the order topology, cheers!
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– Quantum Chill
Mar 12 at 14:55
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I don't know anything about residues, but one thing you might consider is the following: given a particular set how many different sets can you get to using the residue, complement, interior, and closure operators? I believe with closure, complement, and interior the answer is 14. Maybe you can get all sets if you add residue?
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– Robert Thingum
Mar 7 at 1:07
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@RobertThingum I actually did something similar with the boundary, building sets only using A, the boundary operator, and union. There I got 13 distinct sets (not counting the empty set), so I might try the same with the residue operator. Thanks a bunch for the suggestion!
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– Quantum Chill
Mar 7 at 13:49
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In the event you can get every set (doubtful). You should potentially design a structure with operations that act as residue, interior, etc. that is in all but the literal sense a topological space. The study of frames and locales is somewhat like this.
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– Robert Thingum
Mar 7 at 13:54
1
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A closely related concept is defined in: Ronald Brown, Topology and Groupoids, Ex. 2.2.8. What is usually called the boundary of a subset is here called its frontier, and what you call the residue of a subset is what here would be called the boundary of its complement: "For any subset $A$ of a topological space $X$, define [...] $operatornameBdA$ (the boundary of $A$) [...] as follows: [...] $operatornameBdA=A setminusoperatornameIntA$ [...]" There follows a list of identities to be proved, including some involving $operatornameBdA$.
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– Calum Gilhooley
Mar 11 at 22:22
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@CalumGilhooley Thanks for the reference! I actually just defined the above concept of boundary myself, so far I'm calling it the perimeter of $A$, and I found that I can represent my notion of boundary of $A$ as a disjoint union of the residue and perimeter. These two sets seem to be dual to each other in a way, as the perimeter is empty for open sets and the residue is empty for closed sets.
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– Quantum Chill
Mar 12 at 14:49