General topology - challenging problem.
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
While revising my knowledge in general topology, I tackle various difficult problems. I came across one particular problem at which I got completely stuck. Below is the problem:
Assume: on $mathbbR^2$ there are the set of countably many equilateral triangles $ T_i_i=1^+infty$ and the set of countably many straight lines $ L_i_i=1^+infty$.
Prove: there exists a point that is neither equidistant to any pair of straigth lines from $ L_i_i=1^+infty$ nor a vertex of any triangle from $ T_i_i=1^+infty$.
Any help will be highly appreciated!
general-topology contest-math low-dimensional-topology
asked Aug 19 at 9:45
MathTripos
306
 |Â
show 4 more comments
up vote
1
down vote
favorite
While revising my knowledge in general topology, I tackle various difficult problems. I came across one particular problem at which I got completely stuck. Below is the problem:
Assume: on $mathbbR^2$ there are the set of countably many equilateral triangles $ T_i_i=1^+infty$ and the set of countably many straight lines $ L_i_i=1^+infty$.
Prove: there exists a point that is neither equidistant to any pair of straigth lines from $ L_i_i=1^+infty$ nor a vertex of any triangle from $ T_i_i=1^+infty$.
Any help will be highly appreciated!
general-topology contest-math low-dimensional-topology
asked Aug 19 at 9:45
MathTripos
306
Why did my post obtain 2 negative remarks? I am quite baffled...
– MathTripos
Aug 19 at 9:47
1
Quite likely because you forgot to include some thoughts of your own on the question. Also, if this is from a contest, you should tell us whether it's still ongoing.
– Arnaud Mortier
Aug 19 at 9:48
2
I usually don't vote down for that, but I also understand why some people do. The question is really nice, and I know the solution. But without any background (where did you see the problem, why is it interesting to you, is it a homework?) and any sign of own efforts, I feel quite unmotivated to write up the solution.
– A. Pongrácz
Aug 19 at 9:51
@Arnaud Mortier this question did not appear in any contest(unless it was covered during USSR). I took this question from old textbook that contains quite interesting question (its cover is however damaged so I can't tell the author of this question).
– MathTripos
Aug 19 at 9:56
@A.Pongracz this question is not any sort of homework (in UK now there is summer vacation). Below I will provide, hopefully, thorough responses to your concerns.
– MathTripos
Aug 19 at 9:59
 |Â
show 4 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
While revising my knowledge in general topology, I tackle various difficult problems. I came across one particular problem at which I got completely stuck. Below is the problem:
Assume: on $mathbbR^2$ there are the set of countably many equilateral triangles $ T_i_i=1^+infty$ and the set of countably many straight lines $ L_i_i=1^+infty$.
Prove: there exists a point that is neither equidistant to any pair of straigth lines from $ L_i_i=1^+infty$ nor a vertex of any triangle from $ T_i_i=1^+infty$.
Any help will be highly appreciated!
general-topology contest-math low-dimensional-topology
asked Aug 19 at 9:45
MathTripos
306
While revising my knowledge in general topology, I tackle various difficult problems. I came across one particular problem at which I got completely stuck. Below is the problem:
Assume: on $mathbbR^2$ there are the set of countably many equilateral triangles $ T_i_i=1^+infty$ and the set of countably many straight lines $ L_i_i=1^+infty$.
Prove: there exists a point that is neither equidistant to any pair of straigth lines from $ L_i_i=1^+infty$ nor a vertex of any triangle from $ T_i_i=1^+infty$.
Any help will be highly appreciated!
general-topology contest-math low-dimensional-topology
asked Aug 19 at 9:45
MathTripos
306
asked Aug 19 at 9:45
MathTripos
306
asked Aug 19 at 9:45
MathTripos
306
asked Aug 19 at 9:45
MathTripos
306
306
Why did my post obtain 2 negative remarks? I am quite baffled...
– MathTripos
Aug 19 at 9:47
1
Quite likely because you forgot to include some thoughts of your own on the question. Also, if this is from a contest, you should tell us whether it's still ongoing.
– Arnaud Mortier
Aug 19 at 9:48
2
I usually don't vote down for that, but I also understand why some people do. The question is really nice, and I know the solution. But without any background (where did you see the problem, why is it interesting to you, is it a homework?) and any sign of own efforts, I feel quite unmotivated to write up the solution.
– A. Pongrácz
Aug 19 at 9:51
@Arnaud Mortier this question did not appear in any contest(unless it was covered during USSR). I took this question from old textbook that contains quite interesting question (its cover is however damaged so I can't tell the author of this question).
– MathTripos
Aug 19 at 9:56
@A.Pongracz this question is not any sort of homework (in UK now there is summer vacation). Below I will provide, hopefully, thorough responses to your concerns.
– MathTripos
Aug 19 at 9:59
 |Â
show 4 more comments
Why did my post obtain 2 negative remarks? I am quite baffled...
– MathTripos
Aug 19 at 9:47
1
Quite likely because you forgot to include some thoughts of your own on the question. Also, if this is from a contest, you should tell us whether it's still ongoing.
– Arnaud Mortier
Aug 19 at 9:48
2
I usually don't vote down for that, but I also understand why some people do. The question is really nice, and I know the solution. But without any background (where did you see the problem, why is it interesting to you, is it a homework?) and any sign of own efforts, I feel quite unmotivated to write up the solution.
– A. Pongrácz
Aug 19 at 9:51
@Arnaud Mortier this question did not appear in any contest(unless it was covered during USSR). I took this question from old textbook that contains quite interesting question (its cover is however damaged so I can't tell the author of this question).
– MathTripos
Aug 19 at 9:56
@A.Pongracz this question is not any sort of homework (in UK now there is summer vacation). Below I will provide, hopefully, thorough responses to your concerns.
– MathTripos
Aug 19 at 9:59
Why did my post obtain 2 negative remarks? I am quite baffled...
– MathTripos
Aug 19 at 9:47
Why did my post obtain 2 negative remarks? I am quite baffled...
– MathTripos
Aug 19 at 9:47
1
1
Quite likely because you forgot to include some thoughts of your own on the question. Also, if this is from a contest, you should tell us whether it's still ongoing.
– Arnaud Mortier
Aug 19 at 9:48
Quite likely because you forgot to include some thoughts of your own on the question. Also, if this is from a contest, you should tell us whether it's still ongoing.
– Arnaud Mortier
Aug 19 at 9:48
2
2
I usually don't vote down for that, but I also understand why some people do. The question is really nice, and I know the solution. But without any background (where did you see the problem, why is it interesting to you, is it a homework?) and any sign of own efforts, I feel quite unmotivated to write up the solution.
– A. Pongrácz
Aug 19 at 9:51
I usually don't vote down for that, but I also understand why some people do. The question is really nice, and I know the solution. But without any background (where did you see the problem, why is it interesting to you, is it a homework?) and any sign of own efforts, I feel quite unmotivated to write up the solution.
– A. Pongrácz
Aug 19 at 9:51
@Arnaud Mortier this question did not appear in any contest(unless it was covered during USSR). I took this question from old textbook that contains quite interesting question (its cover is however damaged so I can't tell the author of this question).
– MathTripos
Aug 19 at 9:56
@Arnaud Mortier this question did not appear in any contest(unless it was covered during USSR). I took this question from old textbook that contains quite interesting question (its cover is however damaged so I can't tell the author of this question).
– MathTripos
Aug 19 at 9:56
@A.Pongracz this question is not any sort of homework (in UK now there is summer vacation). Below I will provide, hopefully, thorough responses to your concerns.
– MathTripos
Aug 19 at 9:59
@A.Pongracz this question is not any sort of homework (in UK now there is summer vacation). Below I will provide, hopefully, thorough responses to your concerns.
– MathTripos
Aug 19 at 9:59
 |Â
show 4 more comments
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Given two lines, the points equidistant to them is a pair of perperdicular lines. As there are countably many given lines, the set of all these pairs of perpendicular lines is also a countable collection of lines.
Hint: pick a line that is different from all of these, and see what happens on it.
(Why can you pick such a line? What is the cardinality of the points on this line? What is the cardinality of points that you cannot pick on this line?)
I would say that this problem has nothing to do with topology. It is set theory.
answered Aug 19 at 10:07
A. Pongrácz
3,927625
Set theory and some geometry, but definitely no topology.
– Wojowu
Aug 19 at 10:21
@A.Pongracz I see the solution. Thank you very much for your help!
– MathTripos
Aug 19 at 10:21
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Given two lines, the points equidistant to them is a pair of perperdicular lines. As there are countably many given lines, the set of all these pairs of perpendicular lines is also a countable collection of lines.
Hint: pick a line that is different from all of these, and see what happens on it.
(Why can you pick such a line? What is the cardinality of the points on this line? What is the cardinality of points that you cannot pick on this line?)
I would say that this problem has nothing to do with topology. It is set theory.
answered Aug 19 at 10:07
A. Pongrácz
3,927625
Set theory and some geometry, but definitely no topology.
– Wojowu
Aug 19 at 10:21
@A.Pongracz I see the solution. Thank you very much for your help!
– MathTripos
Aug 19 at 10:21
add a comment |Â
up vote
1
down vote
accepted
Given two lines, the points equidistant to them is a pair of perperdicular lines. As there are countably many given lines, the set of all these pairs of perpendicular lines is also a countable collection of lines.
Hint: pick a line that is different from all of these, and see what happens on it.
(Why can you pick such a line? What is the cardinality of the points on this line? What is the cardinality of points that you cannot pick on this line?)
I would say that this problem has nothing to do with topology. It is set theory.
answered Aug 19 at 10:07
A. Pongrácz
3,927625
Set theory and some geometry, but definitely no topology.
– Wojowu
Aug 19 at 10:21
@A.Pongracz I see the solution. Thank you very much for your help!
– MathTripos
Aug 19 at 10:21
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Given two lines, the points equidistant to them is a pair of perperdicular lines. As there are countably many given lines, the set of all these pairs of perpendicular lines is also a countable collection of lines.
Hint: pick a line that is different from all of these, and see what happens on it.
(Why can you pick such a line? What is the cardinality of the points on this line? What is the cardinality of points that you cannot pick on this line?)
I would say that this problem has nothing to do with topology. It is set theory.
answered Aug 19 at 10:07
A. Pongrácz
3,927625
Given two lines, the points equidistant to them is a pair of perperdicular lines. As there are countably many given lines, the set of all these pairs of perpendicular lines is also a countable collection of lines.
Hint: pick a line that is different from all of these, and see what happens on it.
(Why can you pick such a line? What is the cardinality of the points on this line? What is the cardinality of points that you cannot pick on this line?)
I would say that this problem has nothing to do with topology. It is set theory.
answered Aug 19 at 10:07
A. Pongrácz
3,927625
edited Aug 19 at 10:13
answered Aug 19 at 10:07
A. Pongrácz
3,927625
answered Aug 19 at 10:07
A. Pongrácz
3,927625
answered Aug 19 at 10:07
A. Pongrácz
3,927625
3,927625
Set theory and some geometry, but definitely no topology.
– Wojowu
Aug 19 at 10:21
@A.Pongracz I see the solution. Thank you very much for your help!
– MathTripos
Aug 19 at 10:21
add a comment |Â
Set theory and some geometry, but definitely no topology.
– Wojowu
Aug 19 at 10:21
@A.Pongracz I see the solution. Thank you very much for your help!
– MathTripos
Aug 19 at 10:21
Set theory and some geometry, but definitely no topology.
– Wojowu
Aug 19 at 10:21
Set theory and some geometry, but definitely no topology.
– Wojowu
Aug 19 at 10:21
@A.Pongracz I see the solution. Thank you very much for your help!
– MathTripos
Aug 19 at 10:21
@A.Pongracz I see the solution. Thank you very much for your help!
– MathTripos
Aug 19 at 10:21
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%2f2887539%2fgeneral-topology-challenging-problem%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
Why did my post obtain 2 negative remarks? I am quite baffled...
– MathTripos
Aug 19 at 9:47
1
Quite likely because you forgot to include some thoughts of your own on the question. Also, if this is from a contest, you should tell us whether it's still ongoing.
– Arnaud Mortier
Aug 19 at 9:48
2
I usually don't vote down for that, but I also understand why some people do. The question is really nice, and I know the solution. But without any background (where did you see the problem, why is it interesting to you, is it a homework?) and any sign of own efforts, I feel quite unmotivated to write up the solution.
– A. Pongrácz
Aug 19 at 9:51
@Arnaud Mortier this question did not appear in any contest(unless it was covered during USSR). I took this question from old textbook that contains quite interesting question (its cover is however damaged so I can't tell the author of this question).
– MathTripos
Aug 19 at 9:56
@A.Pongracz this question is not any sort of homework (in UK now there is summer vacation). Below I will provide, hopefully, thorough responses to your concerns.
– MathTripos
Aug 19 at 9:59