Mixing
One-Sided Notion of Topological Closure
Clash Royale CLAN TAG#URR8PPP
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Has the following notion already been studied? I am really stuck on a problem that seems to require some general results concerning this:
Let $A$ be a subspace of $mathbbR$. Define an operation called 'left closure' to be $lcl(A)$, that sends $A$ to the subset
$lbrace x in mathbbR : text there exist y_n in A text s.t. y_n leq x$ $forall$ $n text and y_n rightarrow x rbrace$
So this is like the left-sided closure of $A$; it contains all the points that $A$ 'converges to' from the left. We can define $rcl(A)$ similarly. I am wondering what the best way to think about these objects is. There seem like too many choices. In my particular situation, I am dealing with topologically unpleasant objects (uncountable unions of pw-disjoint Cantor Sets), so anything that is well suited to such applications is preferable.
To ask more of a question than a reference request, what can be said about infinite unions of Cantor sets, either countably or uncountable? Anything that falls out of the definition?
The problem that I am especially interested in is this: If $C$ is the union of an uncountable collection of pw disjoint Cantor Sets $C_alpha$, is it possible to apply an order to the indices taking values from $[0,1]$ so that the union of $C_alpha, 0 leq alpha leq alpha_0$ is right-closed in $C$ for any fixed $alpha_0$? That is to say that any collection of sets 'to the left' of some $C_alpha_0$ doesn't accumulate to one of its points from the right.
Thanks! I tagged some topics that seem most likely to contribute, but if anyone has other suggestions, then that would also be helpful.
general-topology monotone-functions measurable-functions semicontinuous-functions sorgenfrey-line
asked Aug 6 at 23:08
John Samples
1,011416
add a comment |Â
up vote
2
down vote
favorite
Has the following notion already been studied? I am really stuck on a problem that seems to require some general results concerning this:
Let $A$ be a subspace of $mathbbR$. Define an operation called 'left closure' to be $lcl(A)$, that sends $A$ to the subset
$lbrace x in mathbbR : text there exist y_n in A text s.t. y_n leq x$ $forall$ $n text and y_n rightarrow x rbrace$
So this is like the left-sided closure of $A$; it contains all the points that $A$ 'converges to' from the left. We can define $rcl(A)$ similarly. I am wondering what the best way to think about these objects is. There seem like too many choices. In my particular situation, I am dealing with topologically unpleasant objects (uncountable unions of pw-disjoint Cantor Sets), so anything that is well suited to such applications is preferable.
To ask more of a question than a reference request, what can be said about infinite unions of Cantor sets, either countably or uncountable? Anything that falls out of the definition?
The problem that I am especially interested in is this: If $C$ is the union of an uncountable collection of pw disjoint Cantor Sets $C_alpha$, is it possible to apply an order to the indices taking values from $[0,1]$ so that the union of $C_alpha, 0 leq alpha leq alpha_0$ is right-closed in $C$ for any fixed $alpha_0$? That is to say that any collection of sets 'to the left' of some $C_alpha_0$ doesn't accumulate to one of its points from the right.
Thanks! I tagged some topics that seem most likely to contribute, but if anyone has other suggestions, then that would also be helpful.
general-topology monotone-functions measurable-functions semicontinuous-functions sorgenfrey-line
asked Aug 6 at 23:08
John Samples
1,011416
I don't understand the question. What do you mean by "what can be said about infinite unions of Cantor sets"? What are "these objects"? What would "fall out" of which definition?
– tomasz
Aug 7 at 0:08
What are some rules for how families of Cantor Sets can left-converge onto each other? For example, is it possible to place an order on an uncountable collection of Cantor Sets so that only predecessors can converge from the left onto a point of some fixed Cantor Set? For example, see this problem. math.stackexchange.com/questions/2872483/…
– John Samples
Aug 7 at 2:22
Yeah. I still don't get it. If you want anyone to answer the question, I think you need to make it more precise.
– tomasz
Aug 7 at 12:11
Suppose instead of Cantor set, you say "Uncountable nowhere-dense perfect set". Do you really want $C$ to be a collection of isometric copies of "the" Cantor set?
– DanielWainfleet
Aug 7 at 16:03
No no, not isometric copies. Just topological homeomorphs.
– John Samples
Aug 7 at 20:10
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Has the following notion already been studied? I am really stuck on a problem that seems to require some general results concerning this:
Let $A$ be a subspace of $mathbbR$. Define an operation called 'left closure' to be $lcl(A)$, that sends $A$ to the subset
$lbrace x in mathbbR : text there exist y_n in A text s.t. y_n leq x$ $forall$ $n text and y_n rightarrow x rbrace$
So this is like the left-sided closure of $A$; it contains all the points that $A$ 'converges to' from the left. We can define $rcl(A)$ similarly. I am wondering what the best way to think about these objects is. There seem like too many choices. In my particular situation, I am dealing with topologically unpleasant objects (uncountable unions of pw-disjoint Cantor Sets), so anything that is well suited to such applications is preferable.
To ask more of a question than a reference request, what can be said about infinite unions of Cantor sets, either countably or uncountable? Anything that falls out of the definition?
The problem that I am especially interested in is this: If $C$ is the union of an uncountable collection of pw disjoint Cantor Sets $C_alpha$, is it possible to apply an order to the indices taking values from $[0,1]$ so that the union of $C_alpha, 0 leq alpha leq alpha_0$ is right-closed in $C$ for any fixed $alpha_0$? That is to say that any collection of sets 'to the left' of some $C_alpha_0$ doesn't accumulate to one of its points from the right.
Thanks! I tagged some topics that seem most likely to contribute, but if anyone has other suggestions, then that would also be helpful.
general-topology monotone-functions measurable-functions semicontinuous-functions sorgenfrey-line
asked Aug 6 at 23:08
John Samples
1,011416
Has the following notion already been studied? I am really stuck on a problem that seems to require some general results concerning this:
Let $A$ be a subspace of $mathbbR$. Define an operation called 'left closure' to be $lcl(A)$, that sends $A$ to the subset
$lbrace x in mathbbR : text there exist y_n in A text s.t. y_n leq x$ $forall$ $n text and y_n rightarrow x rbrace$
So this is like the left-sided closure of $A$; it contains all the points that $A$ 'converges to' from the left. We can define $rcl(A)$ similarly. I am wondering what the best way to think about these objects is. There seem like too many choices. In my particular situation, I am dealing with topologically unpleasant objects (uncountable unions of pw-disjoint Cantor Sets), so anything that is well suited to such applications is preferable.
To ask more of a question than a reference request, what can be said about infinite unions of Cantor sets, either countably or uncountable? Anything that falls out of the definition?
The problem that I am especially interested in is this: If $C$ is the union of an uncountable collection of pw disjoint Cantor Sets $C_alpha$, is it possible to apply an order to the indices taking values from $[0,1]$ so that the union of $C_alpha, 0 leq alpha leq alpha_0$ is right-closed in $C$ for any fixed $alpha_0$? That is to say that any collection of sets 'to the left' of some $C_alpha_0$ doesn't accumulate to one of its points from the right.
Thanks! I tagged some topics that seem most likely to contribute, but if anyone has other suggestions, then that would also be helpful.
general-topology monotone-functions measurable-functions semicontinuous-functions sorgenfrey-line
asked Aug 6 at 23:08
John Samples
1,011416
edited Aug 7 at 6:07
asked Aug 6 at 23:08
John Samples
1,011416
asked Aug 6 at 23:08
John Samples
1,011416
asked Aug 6 at 23:08
John Samples
1,011416
1,011416
I don't understand the question. What do you mean by "what can be said about infinite unions of Cantor sets"? What are "these objects"? What would "fall out" of which definition?
– tomasz
Aug 7 at 0:08
What are some rules for how families of Cantor Sets can left-converge onto each other? For example, is it possible to place an order on an uncountable collection of Cantor Sets so that only predecessors can converge from the left onto a point of some fixed Cantor Set? For example, see this problem. math.stackexchange.com/questions/2872483/…
– John Samples
Aug 7 at 2:22
Yeah. I still don't get it. If you want anyone to answer the question, I think you need to make it more precise.
– tomasz
Aug 7 at 12:11
Suppose instead of Cantor set, you say "Uncountable nowhere-dense perfect set". Do you really want $C$ to be a collection of isometric copies of "the" Cantor set?
– DanielWainfleet
Aug 7 at 16:03
No no, not isometric copies. Just topological homeomorphs.
– John Samples
Aug 7 at 20:10
add a comment |Â
I don't understand the question. What do you mean by "what can be said about infinite unions of Cantor sets"? What are "these objects"? What would "fall out" of which definition?
– tomasz
Aug 7 at 0:08
What are some rules for how families of Cantor Sets can left-converge onto each other? For example, is it possible to place an order on an uncountable collection of Cantor Sets so that only predecessors can converge from the left onto a point of some fixed Cantor Set? For example, see this problem. math.stackexchange.com/questions/2872483/…
– John Samples
Aug 7 at 2:22
Yeah. I still don't get it. If you want anyone to answer the question, I think you need to make it more precise.
– tomasz
Aug 7 at 12:11
Suppose instead of Cantor set, you say "Uncountable nowhere-dense perfect set". Do you really want $C$ to be a collection of isometric copies of "the" Cantor set?
– DanielWainfleet
Aug 7 at 16:03
No no, not isometric copies. Just topological homeomorphs.
– John Samples
Aug 7 at 20:10
I don't understand the question. What do you mean by "what can be said about infinite unions of Cantor sets"? What are "these objects"? What would "fall out" of which definition?
– tomasz
Aug 7 at 0:08
I don't understand the question. What do you mean by "what can be said about infinite unions of Cantor sets"? What are "these objects"? What would "fall out" of which definition?
– tomasz
Aug 7 at 0:08
What are some rules for how families of Cantor Sets can left-converge onto each other? For example, is it possible to place an order on an uncountable collection of Cantor Sets so that only predecessors can converge from the left onto a point of some fixed Cantor Set? For example, see this problem. math.stackexchange.com/questions/2872483/…
– John Samples
Aug 7 at 2:22
What are some rules for how families of Cantor Sets can left-converge onto each other? For example, is it possible to place an order on an uncountable collection of Cantor Sets so that only predecessors can converge from the left onto a point of some fixed Cantor Set? For example, see this problem. math.stackexchange.com/questions/2872483/…
– John Samples
Aug 7 at 2:22
Yeah. I still don't get it. If you want anyone to answer the question, I think you need to make it more precise.
– tomasz
Aug 7 at 12:11
Yeah. I still don't get it. If you want anyone to answer the question, I think you need to make it more precise.
– tomasz
Aug 7 at 12:11
Suppose instead of Cantor set, you say "Uncountable nowhere-dense perfect set". Do you really want $C$ to be a collection of isometric copies of "the" Cantor set?
– DanielWainfleet
Aug 7 at 16:03
Suppose instead of Cantor set, you say "Uncountable nowhere-dense perfect set". Do you really want $C$ to be a collection of isometric copies of "the" Cantor set?
– DanielWainfleet
Aug 7 at 16:03
No no, not isometric copies. Just topological homeomorphs.
– John Samples
Aug 7 at 20:10
No no, not isometric copies. Just topological homeomorphs.
– John Samples
Aug 7 at 20:10
add a comment |Â
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I don't understand the question. What do you mean by "what can be said about infinite unions of Cantor sets"? What are "these objects"? What would "fall out" of which definition?
– tomasz
Aug 7 at 0:08
What are some rules for how families of Cantor Sets can left-converge onto each other? For example, is it possible to place an order on an uncountable collection of Cantor Sets so that only predecessors can converge from the left onto a point of some fixed Cantor Set? For example, see this problem. math.stackexchange.com/questions/2872483/…
– John Samples
Aug 7 at 2:22
Yeah. I still don't get it. If you want anyone to answer the question, I think you need to make it more precise.
– tomasz
Aug 7 at 12:11
Suppose instead of Cantor set, you say "Uncountable nowhere-dense perfect set". Do you really want $C$ to be a collection of isometric copies of "the" Cantor set?
– DanielWainfleet
Aug 7 at 16:03
No no, not isometric copies. Just topological homeomorphs.
– John Samples
Aug 7 at 20:10