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Difference Between Zero-Dimensional and Totally Separated in the Plane
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In Michael Coornaert's book 'Topological Dimension and Dynamical Systems' he mentions the existence of a set $X$ in the plane that is totally separated, but not zero-dimensional. He says it is due to Sierpinski, but the paper he cites is in French. The example doesn't seem to be in Steen & Seebach; is anyone familiar?
Here, 'totally separated' means that the quasicomponents of $X$ are points. Equivalently, for any $x, y in X$ there is a separation of $X$ into disjoint open sets $U, V$ such that $x in U$ and $y in V$.
The Sierpinski paper is here: https://eudml.org/doc/212954
A zero-dimensional space in this context can equivalently mean a space with a clopen basis, a space all of whose points have a clopen basis, or a space with arbitrarily fine covers by disjoint open sets.
EDIT: David Hartley below has described Sierpinski's space satisfying the desired requirements. I've attached a (bad) drawing.
general-topology examples-counterexamples connectedness plane-geometry dimension-theory
asked Jul 29 at 21:27
John Samples
1,011415
 |Â
show 3 more comments
up vote
3
down vote
favorite
In Michael Coornaert's book 'Topological Dimension and Dynamical Systems' he mentions the existence of a set $X$ in the plane that is totally separated, but not zero-dimensional. He says it is due to Sierpinski, but the paper he cites is in French. The example doesn't seem to be in Steen & Seebach; is anyone familiar?
Here, 'totally separated' means that the quasicomponents of $X$ are points. Equivalently, for any $x, y in X$ there is a separation of $X$ into disjoint open sets $U, V$ such that $x in U$ and $y in V$.
The Sierpinski paper is here: https://eudml.org/doc/212954
A zero-dimensional space in this context can equivalently mean a space with a clopen basis, a space all of whose points have a clopen basis, or a space with arbitrarily fine covers by disjoint open sets.
EDIT: David Hartley below has described Sierpinski's space satisfying the desired requirements. I've attached a (bad) drawing.
general-topology examples-counterexamples connectedness plane-geometry dimension-theory
asked Jul 29 at 21:27
John Samples
1,011415
1
What notion of dimension are you using?
– Lorenzo
Jul 29 at 21:35
Since we are in the separable metric case, any of the standard definitions will do. Covering, small ind, big ind or scattered.
– John Samples
Jul 29 at 21:43
I was wondering if the Cantor set counted as an example.
– Lorenzo
Jul 29 at 21:47
1
But can you also provide, for sake of completeness, an explicit description for zero dimensionality?
– Berci
Jul 29 at 21:48
1
Can you post a link to the paper by Sierpinski?
– Rob Arthan
Jul 29 at 21:53
 |Â
show 3 more comments
up vote
3
down vote
favorite
up vote
3
down vote
favorite
In Michael Coornaert's book 'Topological Dimension and Dynamical Systems' he mentions the existence of a set $X$ in the plane that is totally separated, but not zero-dimensional. He says it is due to Sierpinski, but the paper he cites is in French. The example doesn't seem to be in Steen & Seebach; is anyone familiar?
Here, 'totally separated' means that the quasicomponents of $X$ are points. Equivalently, for any $x, y in X$ there is a separation of $X$ into disjoint open sets $U, V$ such that $x in U$ and $y in V$.
The Sierpinski paper is here: https://eudml.org/doc/212954
A zero-dimensional space in this context can equivalently mean a space with a clopen basis, a space all of whose points have a clopen basis, or a space with arbitrarily fine covers by disjoint open sets.
EDIT: David Hartley below has described Sierpinski's space satisfying the desired requirements. I've attached a (bad) drawing.
general-topology examples-counterexamples connectedness plane-geometry dimension-theory
asked Jul 29 at 21:27
John Samples
1,011415
In Michael Coornaert's book 'Topological Dimension and Dynamical Systems' he mentions the existence of a set $X$ in the plane that is totally separated, but not zero-dimensional. He says it is due to Sierpinski, but the paper he cites is in French. The example doesn't seem to be in Steen & Seebach; is anyone familiar?
Here, 'totally separated' means that the quasicomponents of $X$ are points. Equivalently, for any $x, y in X$ there is a separation of $X$ into disjoint open sets $U, V$ such that $x in U$ and $y in V$.
The Sierpinski paper is here: https://eudml.org/doc/212954
A zero-dimensional space in this context can equivalently mean a space with a clopen basis, a space all of whose points have a clopen basis, or a space with arbitrarily fine covers by disjoint open sets.
EDIT: David Hartley below has described Sierpinski's space satisfying the desired requirements. I've attached a (bad) drawing.
general-topology examples-counterexamples connectedness plane-geometry dimension-theory
asked Jul 29 at 21:27
John Samples
1,011415
edited Aug 1 at 3:49
asked Jul 29 at 21:27
John Samples
1,011415
asked Jul 29 at 21:27
John Samples
1,011415
asked Jul 29 at 21:27
John Samples
1,011415
1,011415
1
What notion of dimension are you using?
– Lorenzo
Jul 29 at 21:35
Since we are in the separable metric case, any of the standard definitions will do. Covering, small ind, big ind or scattered.
– John Samples
Jul 29 at 21:43
I was wondering if the Cantor set counted as an example.
– Lorenzo
Jul 29 at 21:47
1
But can you also provide, for sake of completeness, an explicit description for zero dimensionality?
– Berci
Jul 29 at 21:48
1
Can you post a link to the paper by Sierpinski?
– Rob Arthan
Jul 29 at 21:53
 |Â
show 3 more comments
1
What notion of dimension are you using?
– Lorenzo
Jul 29 at 21:35
Since we are in the separable metric case, any of the standard definitions will do. Covering, small ind, big ind or scattered.
– John Samples
Jul 29 at 21:43
I was wondering if the Cantor set counted as an example.
– Lorenzo
Jul 29 at 21:47
1
But can you also provide, for sake of completeness, an explicit description for zero dimensionality?
– Berci
Jul 29 at 21:48
1
Can you post a link to the paper by Sierpinski?
– Rob Arthan
Jul 29 at 21:53
1
1
What notion of dimension are you using?
– Lorenzo
Jul 29 at 21:35
What notion of dimension are you using?
– Lorenzo
Jul 29 at 21:35
Since we are in the separable metric case, any of the standard definitions will do. Covering, small ind, big ind or scattered.
– John Samples
Jul 29 at 21:43
Since we are in the separable metric case, any of the standard definitions will do. Covering, small ind, big ind or scattered.
– John Samples
Jul 29 at 21:43
I was wondering if the Cantor set counted as an example.
– Lorenzo
Jul 29 at 21:47
I was wondering if the Cantor set counted as an example.
– Lorenzo
Jul 29 at 21:47
1
1
But can you also provide, for sake of completeness, an explicit description for zero dimensionality?
– Berci
Jul 29 at 21:48
But can you also provide, for sake of completeness, an explicit description for zero dimensionality?
– Berci
Jul 29 at 21:48
1
1
Can you post a link to the paper by Sierpinski?
– Rob Arthan
Jul 29 at 21:53
Can you post a link to the paper by Sierpinski?
– Rob Arthan
Jul 29 at 21:53
 |Â
show 3 more comments
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
The example in the Sierpinski paper quoted is quite different from his carpet, except it also uses lots of rectangles. Here's a brief sketch of the construction. It uses repeated applications of the following procedure, illustrated on the rectangle with corners at $(pm1,pm1)$. Let $R(2n-1)$ be the rectangle in the top-left quarter bounded by $y=0,y=1,x=-(2^-(2n-2))$ and $x=-(2^-(2n-1))$. Let $R(2n)$ be the matching rectangle
in the bottom-right quarter (rotate 180 about origin). The same procedure,
scaled appropriately, is repeated in each $R(n_1)$ to give rectangles $R(n_1,n_2)$ and again within those, and so on, giving $R(n_1, n_2,..,n_k)$ for any natural k.
Let $S_k$ be the union of all these rectangles with k indices and P be the
intersection of all $S_k$. Let Q be the set containing the centres of all the
defined rectangles and $E = P cup Q$.
$E$ is totally separated but not zero-dimensional. In particular, any clopen
set containing the centre of the outer rectangle has the mid-points of the
top and bottom sides in its closure in $mathbb R^2$ (they are not in $E$) and so has
diameter at least 2. Hence the centre point cannot have a nbhd. basis of clopen sets.
Adding one of those mid-points to $E$ forms $E_1$ which cannot be totally
separated but remains totally disconnected.
answered Jul 30 at 21:58
David Hartley
550148
Should the exponents for the $x$-coordinates be negative?
– John Samples
Jul 31 at 3:19
Ok, this space works! Thanks! The details for the non-zero-dimensionality are a bit unpleasant; do you prefer to post them or should I go ahead?
– John Samples
Jul 31 at 4:21
Yes, they should be negative. I'll change them. (I tried to simplify the original as I'm not much good with Tex, but I slipped up there.)
– David Hartley
Jul 31 at 8:07
I didn't make it clear that this is not my example, just a "translation" of the one in the Sierpinski paper. I'll have a go at the proof if you like, but it will have to wait to tonight.
– David Hartley
Jul 31 at 9:01
if you can I would appreciate it, I have been MSEing on nothing but a tablet this summer (touch screen keyboard). I deserve a new badge haha
– John Samples
Aug 1 at 3:48
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
The example in the Sierpinski paper quoted is quite different from his carpet, except it also uses lots of rectangles. Here's a brief sketch of the construction. It uses repeated applications of the following procedure, illustrated on the rectangle with corners at $(pm1,pm1)$. Let $R(2n-1)$ be the rectangle in the top-left quarter bounded by $y=0,y=1,x=-(2^-(2n-2))$ and $x=-(2^-(2n-1))$. Let $R(2n)$ be the matching rectangle
in the bottom-right quarter (rotate 180 about origin). The same procedure,
scaled appropriately, is repeated in each $R(n_1)$ to give rectangles $R(n_1,n_2)$ and again within those, and so on, giving $R(n_1, n_2,..,n_k)$ for any natural k.
Let $S_k$ be the union of all these rectangles with k indices and P be the
intersection of all $S_k$. Let Q be the set containing the centres of all the
defined rectangles and $E = P cup Q$.
$E$ is totally separated but not zero-dimensional. In particular, any clopen
set containing the centre of the outer rectangle has the mid-points of the
top and bottom sides in its closure in $mathbb R^2$ (they are not in $E$) and so has
diameter at least 2. Hence the centre point cannot have a nbhd. basis of clopen sets.
Adding one of those mid-points to $E$ forms $E_1$ which cannot be totally
separated but remains totally disconnected.
answered Jul 30 at 21:58
David Hartley
550148
Should the exponents for the $x$-coordinates be negative?
– John Samples
Jul 31 at 3:19
Ok, this space works! Thanks! The details for the non-zero-dimensionality are a bit unpleasant; do you prefer to post them or should I go ahead?
– John Samples
Jul 31 at 4:21
Yes, they should be negative. I'll change them. (I tried to simplify the original as I'm not much good with Tex, but I slipped up there.)
– David Hartley
Jul 31 at 8:07
I didn't make it clear that this is not my example, just a "translation" of the one in the Sierpinski paper. I'll have a go at the proof if you like, but it will have to wait to tonight.
– David Hartley
Jul 31 at 9:01
if you can I would appreciate it, I have been MSEing on nothing but a tablet this summer (touch screen keyboard). I deserve a new badge haha
– John Samples
Aug 1 at 3:48
add a comment |Â
up vote
2
down vote
accepted
The example in the Sierpinski paper quoted is quite different from his carpet, except it also uses lots of rectangles. Here's a brief sketch of the construction. It uses repeated applications of the following procedure, illustrated on the rectangle with corners at $(pm1,pm1)$. Let $R(2n-1)$ be the rectangle in the top-left quarter bounded by $y=0,y=1,x=-(2^-(2n-2))$ and $x=-(2^-(2n-1))$. Let $R(2n)$ be the matching rectangle
in the bottom-right quarter (rotate 180 about origin). The same procedure,
scaled appropriately, is repeated in each $R(n_1)$ to give rectangles $R(n_1,n_2)$ and again within those, and so on, giving $R(n_1, n_2,..,n_k)$ for any natural k.
Let $S_k$ be the union of all these rectangles with k indices and P be the
intersection of all $S_k$. Let Q be the set containing the centres of all the
defined rectangles and $E = P cup Q$.
$E$ is totally separated but not zero-dimensional. In particular, any clopen
set containing the centre of the outer rectangle has the mid-points of the
top and bottom sides in its closure in $mathbb R^2$ (they are not in $E$) and so has
diameter at least 2. Hence the centre point cannot have a nbhd. basis of clopen sets.
Adding one of those mid-points to $E$ forms $E_1$ which cannot be totally
separated but remains totally disconnected.
answered Jul 30 at 21:58
David Hartley
550148
Should the exponents for the $x$-coordinates be negative?
– John Samples
Jul 31 at 3:19
Ok, this space works! Thanks! The details for the non-zero-dimensionality are a bit unpleasant; do you prefer to post them or should I go ahead?
– John Samples
Jul 31 at 4:21
Yes, they should be negative. I'll change them. (I tried to simplify the original as I'm not much good with Tex, but I slipped up there.)
– David Hartley
Jul 31 at 8:07
I didn't make it clear that this is not my example, just a "translation" of the one in the Sierpinski paper. I'll have a go at the proof if you like, but it will have to wait to tonight.
– David Hartley
Jul 31 at 9:01
if you can I would appreciate it, I have been MSEing on nothing but a tablet this summer (touch screen keyboard). I deserve a new badge haha
– John Samples
Aug 1 at 3:48
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
The example in the Sierpinski paper quoted is quite different from his carpet, except it also uses lots of rectangles. Here's a brief sketch of the construction. It uses repeated applications of the following procedure, illustrated on the rectangle with corners at $(pm1,pm1)$. Let $R(2n-1)$ be the rectangle in the top-left quarter bounded by $y=0,y=1,x=-(2^-(2n-2))$ and $x=-(2^-(2n-1))$. Let $R(2n)$ be the matching rectangle
in the bottom-right quarter (rotate 180 about origin). The same procedure,
scaled appropriately, is repeated in each $R(n_1)$ to give rectangles $R(n_1,n_2)$ and again within those, and so on, giving $R(n_1, n_2,..,n_k)$ for any natural k.
Let $S_k$ be the union of all these rectangles with k indices and P be the
intersection of all $S_k$. Let Q be the set containing the centres of all the
defined rectangles and $E = P cup Q$.
$E$ is totally separated but not zero-dimensional. In particular, any clopen
set containing the centre of the outer rectangle has the mid-points of the
top and bottom sides in its closure in $mathbb R^2$ (they are not in $E$) and so has
diameter at least 2. Hence the centre point cannot have a nbhd. basis of clopen sets.
Adding one of those mid-points to $E$ forms $E_1$ which cannot be totally
separated but remains totally disconnected.
answered Jul 30 at 21:58
David Hartley
550148
The example in the Sierpinski paper quoted is quite different from his carpet, except it also uses lots of rectangles. Here's a brief sketch of the construction. It uses repeated applications of the following procedure, illustrated on the rectangle with corners at $(pm1,pm1)$. Let $R(2n-1)$ be the rectangle in the top-left quarter bounded by $y=0,y=1,x=-(2^-(2n-2))$ and $x=-(2^-(2n-1))$. Let $R(2n)$ be the matching rectangle
in the bottom-right quarter (rotate 180 about origin). The same procedure,
scaled appropriately, is repeated in each $R(n_1)$ to give rectangles $R(n_1,n_2)$ and again within those, and so on, giving $R(n_1, n_2,..,n_k)$ for any natural k.
Let $S_k$ be the union of all these rectangles with k indices and P be the
intersection of all $S_k$. Let Q be the set containing the centres of all the
defined rectangles and $E = P cup Q$.
$E$ is totally separated but not zero-dimensional. In particular, any clopen
set containing the centre of the outer rectangle has the mid-points of the
top and bottom sides in its closure in $mathbb R^2$ (they are not in $E$) and so has
diameter at least 2. Hence the centre point cannot have a nbhd. basis of clopen sets.
Adding one of those mid-points to $E$ forms $E_1$ which cannot be totally
separated but remains totally disconnected.
answered Jul 30 at 21:58
David Hartley
550148
edited Jul 31 at 9:05
answered Jul 30 at 21:58
David Hartley
550148
answered Jul 30 at 21:58
David Hartley
550148
answered Jul 30 at 21:58
David Hartley
550148
550148
Should the exponents for the $x$-coordinates be negative?
– John Samples
Jul 31 at 3:19
Ok, this space works! Thanks! The details for the non-zero-dimensionality are a bit unpleasant; do you prefer to post them or should I go ahead?
– John Samples
Jul 31 at 4:21
Yes, they should be negative. I'll change them. (I tried to simplify the original as I'm not much good with Tex, but I slipped up there.)
– David Hartley
Jul 31 at 8:07
I didn't make it clear that this is not my example, just a "translation" of the one in the Sierpinski paper. I'll have a go at the proof if you like, but it will have to wait to tonight.
– David Hartley
Jul 31 at 9:01
if you can I would appreciate it, I have been MSEing on nothing but a tablet this summer (touch screen keyboard). I deserve a new badge haha
– John Samples
Aug 1 at 3:48
add a comment |Â
Should the exponents for the $x$-coordinates be negative?
– John Samples
Jul 31 at 3:19
Ok, this space works! Thanks! The details for the non-zero-dimensionality are a bit unpleasant; do you prefer to post them or should I go ahead?
– John Samples
Jul 31 at 4:21
Yes, they should be negative. I'll change them. (I tried to simplify the original as I'm not much good with Tex, but I slipped up there.)
– David Hartley
Jul 31 at 8:07
I didn't make it clear that this is not my example, just a "translation" of the one in the Sierpinski paper. I'll have a go at the proof if you like, but it will have to wait to tonight.
– David Hartley
Jul 31 at 9:01
if you can I would appreciate it, I have been MSEing on nothing but a tablet this summer (touch screen keyboard). I deserve a new badge haha
– John Samples
Aug 1 at 3:48
Should the exponents for the $x$-coordinates be negative?
– John Samples
Jul 31 at 3:19
Should the exponents for the $x$-coordinates be negative?
– John Samples
Jul 31 at 3:19
Ok, this space works! Thanks! The details for the non-zero-dimensionality are a bit unpleasant; do you prefer to post them or should I go ahead?
– John Samples
Jul 31 at 4:21
Ok, this space works! Thanks! The details for the non-zero-dimensionality are a bit unpleasant; do you prefer to post them or should I go ahead?
– John Samples
Jul 31 at 4:21
Yes, they should be negative. I'll change them. (I tried to simplify the original as I'm not much good with Tex, but I slipped up there.)
– David Hartley
Jul 31 at 8:07
Yes, they should be negative. I'll change them. (I tried to simplify the original as I'm not much good with Tex, but I slipped up there.)
– David Hartley
Jul 31 at 8:07
I didn't make it clear that this is not my example, just a "translation" of the one in the Sierpinski paper. I'll have a go at the proof if you like, but it will have to wait to tonight.
– David Hartley
Jul 31 at 9:01
I didn't make it clear that this is not my example, just a "translation" of the one in the Sierpinski paper. I'll have a go at the proof if you like, but it will have to wait to tonight.
– David Hartley
Jul 31 at 9:01
if you can I would appreciate it, I have been MSEing on nothing but a tablet this summer (touch screen keyboard). I deserve a new badge haha
– John Samples
Aug 1 at 3:48
if you can I would appreciate it, I have been MSEing on nothing but a tablet this summer (touch screen keyboard). I deserve a new badge haha
– John Samples
Aug 1 at 3:48
add a comment |Â
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1
What notion of dimension are you using?
– Lorenzo
Jul 29 at 21:35
Since we are in the separable metric case, any of the standard definitions will do. Covering, small ind, big ind or scattered.
– John Samples
Jul 29 at 21:43
I was wondering if the Cantor set counted as an example.
– Lorenzo
Jul 29 at 21:47
1
But can you also provide, for sake of completeness, an explicit description for zero dimensionality?
– Berci
Jul 29 at 21:48
1
Can you post a link to the paper by Sierpinski?
– Rob Arthan
Jul 29 at 21:53