Mixing
Construction of an Open, Dense, Connected Set in the Plane
Clash Royale CLAN TAG#URR8PPP
up vote
3
down vote
favorite
I'm stumped with the following problem.
Let $varepsilon>0$ be given. Prove that there exists an open, dense, and connected set $Gsubset mathbbR^2$ such that $m_2(G)<varepsilon$, where $m_2$ is the Lebesgue measure on $mathbbR^2$.
My thoughts: I'm thinking that I need to use some sort of construction with a Cantor-like set in $mathbbR^2$ and then take a set complement. However, I haven't worked with Cantor sets outside of $mathbbR$, so I'm not sure what constitutes a "Cantor-like set" in higher dimensions (if this is even defined or a valid construction) Is this roughly what I should want to do? Otherwise, I'm not sure where I should start.
Thanks in advance for any help!
real-analysis measure-theory lebesgue-measure
asked 3 hours ago
Sir_Math_Cat
912618
add a comment |Â
up vote
3
down vote
favorite
I'm stumped with the following problem.
Let $varepsilon>0$ be given. Prove that there exists an open, dense, and connected set $Gsubset mathbbR^2$ such that $m_2(G)<varepsilon$, where $m_2$ is the Lebesgue measure on $mathbbR^2$.
My thoughts: I'm thinking that I need to use some sort of construction with a Cantor-like set in $mathbbR^2$ and then take a set complement. However, I haven't worked with Cantor sets outside of $mathbbR$, so I'm not sure what constitutes a "Cantor-like set" in higher dimensions (if this is even defined or a valid construction) Is this roughly what I should want to do? Otherwise, I'm not sure where I should start.
Thanks in advance for any help!
real-analysis measure-theory lebesgue-measure
asked 3 hours ago
Sir_Math_Cat
912618
Draw a sombrero containing $mathbbR$. Then draw a bunch of vertical sombreros chosen well.
– T. Bongers
2 hours ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I'm stumped with the following problem.
Let $varepsilon>0$ be given. Prove that there exists an open, dense, and connected set $Gsubset mathbbR^2$ such that $m_2(G)<varepsilon$, where $m_2$ is the Lebesgue measure on $mathbbR^2$.
My thoughts: I'm thinking that I need to use some sort of construction with a Cantor-like set in $mathbbR^2$ and then take a set complement. However, I haven't worked with Cantor sets outside of $mathbbR$, so I'm not sure what constitutes a "Cantor-like set" in higher dimensions (if this is even defined or a valid construction) Is this roughly what I should want to do? Otherwise, I'm not sure where I should start.
Thanks in advance for any help!
real-analysis measure-theory lebesgue-measure
asked 3 hours ago
Sir_Math_Cat
912618
I'm stumped with the following problem.
Let $varepsilon>0$ be given. Prove that there exists an open, dense, and connected set $Gsubset mathbbR^2$ such that $m_2(G)<varepsilon$, where $m_2$ is the Lebesgue measure on $mathbbR^2$.
My thoughts: I'm thinking that I need to use some sort of construction with a Cantor-like set in $mathbbR^2$ and then take a set complement. However, I haven't worked with Cantor sets outside of $mathbbR$, so I'm not sure what constitutes a "Cantor-like set" in higher dimensions (if this is even defined or a valid construction) Is this roughly what I should want to do? Otherwise, I'm not sure where I should start.
Thanks in advance for any help!
real-analysis measure-theory lebesgue-measure
real-analysis measure-theory lebesgue-measure
asked 3 hours ago
Sir_Math_Cat
912618
asked 3 hours ago
Sir_Math_Cat
912618
asked 3 hours ago
Sir_Math_Cat
912618
asked 3 hours ago
Sir_Math_Cat
912618
asked 3 hours ago
Sir_Math_Cat
912618
912618
Draw a sombrero containing $mathbbR$. Then draw a bunch of vertical sombreros chosen well.
– T. Bongers
2 hours ago
add a comment |Â
Draw a sombrero containing $mathbbR$. Then draw a bunch of vertical sombreros chosen well.
– T. Bongers
2 hours ago
Draw a sombrero containing $mathbbR$. Then draw a bunch of vertical sombreros chosen well.
– T. Bongers
2 hours ago
Draw a sombrero containing $mathbbR$. Then draw a bunch of vertical sombreros chosen well.
– T. Bongers
2 hours ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
3
down vote
Here's a quick outline of a "brute force" construction:
One way to make a dense open set is to enumerate the points both of whose coordinates are rationals (more generally, in $mathbbR^n$ we want to enumerate $mathbbQ^n$) as $(q_i)_iinmathbbN$, and then put an open ball $B_i$ around each $q_i$. Since the rationals are dense, the open set $B=bigcup B_i$ will be dense. Now, do you see a way to pick balls so that $B$ has "small" measure?
Now the result of the above won't be connected (exercise). So we need to make it connected. The idea now is to put "bridges" between the open balls we've already drawn - given $B_i, B_j$, fix points $x_i, x_j$ in each ball (say, their centers) and consider some open set $L_i,j$ around the line segment connecting $x_i$ and $x_j$. Do you see how to design these $L_i,j$s so that the sum of their measures is "small"?
A tangential comment (hidden since it contains spoilers):
The construction above can ultimately lead you in the direction of "higher" metric spaces; that is, metric spaces whose "points" are more usually thought of as sets of points. Here's how. In my opinion, the simplest approach to the second bulletpoint above is to look at the set of points whose distance to the given line segment is $<epsilon$ for an appropriate $epsilon$. This kind of "ball around a set" (as opposed to point) is a very useful notion in metric spaces. In fact, we can leave points behind entirely (well, not really) and define a "distance" function on arbitrary sets in a metric space, namely the infimum of the distances between a point in one set and a point in the other set. This isn't a metric in general, but is when we restrict to appropriate sets (exercise: convince yourself that we should restrict attention to the compact sets) and shows up in a number of situations. (Going further afield, it turns out that this isn't the only reasonable metric to put on ("nice") subsets of a metric space, but that's not related to the current problem at all; I just think it's neato.)
(What, tangential comments shouldn't be longer than the actual relevant answer? Nonsense I say!)
answered 2 hours ago
Noah Schweber
115k9143273
add a comment |Â
up vote
0
down vote
Here is an outline. The construction is based on the fact that you can draw an infinite sombrero that only contains finite area under it. To be precise, $int_-infty^infty fracdx1 + x^2 < infty$.
Density is handled by putting vertical versions of this set at carefully chosen points.
Finite area is handled by scaling.
Connectedness is handled by putting a horizontal version to connect all the vertical ones.
answered 2 hours ago
T. Bongers
21.7k54359
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Here's a quick outline of a "brute force" construction:
One way to make a dense open set is to enumerate the points both of whose coordinates are rationals (more generally, in $mathbbR^n$ we want to enumerate $mathbbQ^n$) as $(q_i)_iinmathbbN$, and then put an open ball $B_i$ around each $q_i$. Since the rationals are dense, the open set $B=bigcup B_i$ will be dense. Now, do you see a way to pick balls so that $B$ has "small" measure?
Now the result of the above won't be connected (exercise). So we need to make it connected. The idea now is to put "bridges" between the open balls we've already drawn - given $B_i, B_j$, fix points $x_i, x_j$ in each ball (say, their centers) and consider some open set $L_i,j$ around the line segment connecting $x_i$ and $x_j$. Do you see how to design these $L_i,j$s so that the sum of their measures is "small"?
A tangential comment (hidden since it contains spoilers):
The construction above can ultimately lead you in the direction of "higher" metric spaces; that is, metric spaces whose "points" are more usually thought of as sets of points. Here's how. In my opinion, the simplest approach to the second bulletpoint above is to look at the set of points whose distance to the given line segment is $<epsilon$ for an appropriate $epsilon$. This kind of "ball around a set" (as opposed to point) is a very useful notion in metric spaces. In fact, we can leave points behind entirely (well, not really) and define a "distance" function on arbitrary sets in a metric space, namely the infimum of the distances between a point in one set and a point in the other set. This isn't a metric in general, but is when we restrict to appropriate sets (exercise: convince yourself that we should restrict attention to the compact sets) and shows up in a number of situations. (Going further afield, it turns out that this isn't the only reasonable metric to put on ("nice") subsets of a metric space, but that's not related to the current problem at all; I just think it's neato.)
(What, tangential comments shouldn't be longer than the actual relevant answer? Nonsense I say!)
answered 2 hours ago
Noah Schweber
115k9143273
add a comment |Â
up vote
3
down vote
Here's a quick outline of a "brute force" construction:
One way to make a dense open set is to enumerate the points both of whose coordinates are rationals (more generally, in $mathbbR^n$ we want to enumerate $mathbbQ^n$) as $(q_i)_iinmathbbN$, and then put an open ball $B_i$ around each $q_i$. Since the rationals are dense, the open set $B=bigcup B_i$ will be dense. Now, do you see a way to pick balls so that $B$ has "small" measure?
Now the result of the above won't be connected (exercise). So we need to make it connected. The idea now is to put "bridges" between the open balls we've already drawn - given $B_i, B_j$, fix points $x_i, x_j$ in each ball (say, their centers) and consider some open set $L_i,j$ around the line segment connecting $x_i$ and $x_j$. Do you see how to design these $L_i,j$s so that the sum of their measures is "small"?
A tangential comment (hidden since it contains spoilers):
The construction above can ultimately lead you in the direction of "higher" metric spaces; that is, metric spaces whose "points" are more usually thought of as sets of points. Here's how. In my opinion, the simplest approach to the second bulletpoint above is to look at the set of points whose distance to the given line segment is $<epsilon$ for an appropriate $epsilon$. This kind of "ball around a set" (as opposed to point) is a very useful notion in metric spaces. In fact, we can leave points behind entirely (well, not really) and define a "distance" function on arbitrary sets in a metric space, namely the infimum of the distances between a point in one set and a point in the other set. This isn't a metric in general, but is when we restrict to appropriate sets (exercise: convince yourself that we should restrict attention to the compact sets) and shows up in a number of situations. (Going further afield, it turns out that this isn't the only reasonable metric to put on ("nice") subsets of a metric space, but that's not related to the current problem at all; I just think it's neato.)
(What, tangential comments shouldn't be longer than the actual relevant answer? Nonsense I say!)
answered 2 hours ago
Noah Schweber
115k9143273
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Here's a quick outline of a "brute force" construction:
One way to make a dense open set is to enumerate the points both of whose coordinates are rationals (more generally, in $mathbbR^n$ we want to enumerate $mathbbQ^n$) as $(q_i)_iinmathbbN$, and then put an open ball $B_i$ around each $q_i$. Since the rationals are dense, the open set $B=bigcup B_i$ will be dense. Now, do you see a way to pick balls so that $B$ has "small" measure?
Now the result of the above won't be connected (exercise). So we need to make it connected. The idea now is to put "bridges" between the open balls we've already drawn - given $B_i, B_j$, fix points $x_i, x_j$ in each ball (say, their centers) and consider some open set $L_i,j$ around the line segment connecting $x_i$ and $x_j$. Do you see how to design these $L_i,j$s so that the sum of their measures is "small"?
A tangential comment (hidden since it contains spoilers):
The construction above can ultimately lead you in the direction of "higher" metric spaces; that is, metric spaces whose "points" are more usually thought of as sets of points. Here's how. In my opinion, the simplest approach to the second bulletpoint above is to look at the set of points whose distance to the given line segment is $<epsilon$ for an appropriate $epsilon$. This kind of "ball around a set" (as opposed to point) is a very useful notion in metric spaces. In fact, we can leave points behind entirely (well, not really) and define a "distance" function on arbitrary sets in a metric space, namely the infimum of the distances between a point in one set and a point in the other set. This isn't a metric in general, but is when we restrict to appropriate sets (exercise: convince yourself that we should restrict attention to the compact sets) and shows up in a number of situations. (Going further afield, it turns out that this isn't the only reasonable metric to put on ("nice") subsets of a metric space, but that's not related to the current problem at all; I just think it's neato.)
(What, tangential comments shouldn't be longer than the actual relevant answer? Nonsense I say!)
answered 2 hours ago
Noah Schweber
115k9143273
Here's a quick outline of a "brute force" construction:
One way to make a dense open set is to enumerate the points both of whose coordinates are rationals (more generally, in $mathbbR^n$ we want to enumerate $mathbbQ^n$) as $(q_i)_iinmathbbN$, and then put an open ball $B_i$ around each $q_i$. Since the rationals are dense, the open set $B=bigcup B_i$ will be dense. Now, do you see a way to pick balls so that $B$ has "small" measure?
Now the result of the above won't be connected (exercise). So we need to make it connected. The idea now is to put "bridges" between the open balls we've already drawn - given $B_i, B_j$, fix points $x_i, x_j$ in each ball (say, their centers) and consider some open set $L_i,j$ around the line segment connecting $x_i$ and $x_j$. Do you see how to design these $L_i,j$s so that the sum of their measures is "small"?
A tangential comment (hidden since it contains spoilers):
The construction above can ultimately lead you in the direction of "higher" metric spaces; that is, metric spaces whose "points" are more usually thought of as sets of points. Here's how. In my opinion, the simplest approach to the second bulletpoint above is to look at the set of points whose distance to the given line segment is $<epsilon$ for an appropriate $epsilon$. This kind of "ball around a set" (as opposed to point) is a very useful notion in metric spaces. In fact, we can leave points behind entirely (well, not really) and define a "distance" function on arbitrary sets in a metric space, namely the infimum of the distances between a point in one set and a point in the other set. This isn't a metric in general, but is when we restrict to appropriate sets (exercise: convince yourself that we should restrict attention to the compact sets) and shows up in a number of situations. (Going further afield, it turns out that this isn't the only reasonable metric to put on ("nice") subsets of a metric space, but that's not related to the current problem at all; I just think it's neato.)
(What, tangential comments shouldn't be longer than the actual relevant answer? Nonsense I say!)
answered 2 hours ago
Noah Schweber
115k9143273
edited 19 mins ago
answered 2 hours ago
Noah Schweber
115k9143273
answered 2 hours ago
Noah Schweber
115k9143273
answered 2 hours ago
Noah Schweber
115k9143273
115k9143273
add a comment |Â
add a comment |Â
up vote
0
down vote
Here is an outline. The construction is based on the fact that you can draw an infinite sombrero that only contains finite area under it. To be precise, $int_-infty^infty fracdx1 + x^2 < infty$.
Density is handled by putting vertical versions of this set at carefully chosen points.
Finite area is handled by scaling.
Connectedness is handled by putting a horizontal version to connect all the vertical ones.
answered 2 hours ago
T. Bongers
21.7k54359
add a comment |Â
up vote
0
down vote
Here is an outline. The construction is based on the fact that you can draw an infinite sombrero that only contains finite area under it. To be precise, $int_-infty^infty fracdx1 + x^2 < infty$.
Density is handled by putting vertical versions of this set at carefully chosen points.
Finite area is handled by scaling.
Connectedness is handled by putting a horizontal version to connect all the vertical ones.
answered 2 hours ago
T. Bongers
21.7k54359
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Here is an outline. The construction is based on the fact that you can draw an infinite sombrero that only contains finite area under it. To be precise, $int_-infty^infty fracdx1 + x^2 < infty$.
Density is handled by putting vertical versions of this set at carefully chosen points.
Finite area is handled by scaling.
Connectedness is handled by putting a horizontal version to connect all the vertical ones.
answered 2 hours ago
T. Bongers
21.7k54359
Here is an outline. The construction is based on the fact that you can draw an infinite sombrero that only contains finite area under it. To be precise, $int_-infty^infty fracdx1 + x^2 < infty$.
Density is handled by putting vertical versions of this set at carefully chosen points.
Finite area is handled by scaling.
Connectedness is handled by putting a horizontal version to connect all the vertical ones.
answered 2 hours ago
T. Bongers
21.7k54359
answered 2 hours ago
T. Bongers
21.7k54359
answered 2 hours ago
T. Bongers
21.7k54359
answered 2 hours ago
T. Bongers
21.7k54359
21.7k54359
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2963367%2fconstruction-of-an-open-dense-connected-set-in-the-plane%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Draw a sombrero containing $mathbbR$. Then draw a bunch of vertical sombreros chosen well.
– T. Bongers
2 hours ago