Bicontinuous bijection between line and squareIs there a continuous bijection between an interval $[0,1]$ and a square: $[0,1] times [0,1]$?Bijection between $[0,1]$ and $K$Bijection between measurable setsBijection between open and closed intervalPath From Positive Dedekind Cuts to Reals?Bijection between closed unit interval and $ R$Strong Counterexample to MVT on QStrong Counterexample to MVT on QHow do we know that there are not more numbers than there are names?Direct bijection between $C[0,1]$ and $[0,1]$bijection between the Cantor set and $R$
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Bicontinuous bijection between line and square
Is there a continuous bijection between an interval $[0,1]$ and a square: $[0,1] times [0,1]$?Bijection between $[0,1]$ and $K$Bijection between measurable setsBijection between open and closed intervalPath From Positive Dedekind Cuts to Reals?Bijection between closed unit interval and $ R$Strong Counterexample to MVT on QStrong Counterexample to MVT on QHow do we know that there are not more numbers than there are names?Direct bijection between $C[0,1]$ and $[0,1]$bijection between the Cantor set and $R$
$begingroup$
A few years ago I saw a reference to two articles by two prominent 19th Century mathematicians, whose names I can no longer remember, to constructions of bicontinuous bijections between an interval, maybe open, maybe closed, of the reals, and a square, similarly open or closed, in 2-dim. Euclidean space, which I wouldn't have believed possible. However, I couldn't find any access to the articles. Can anyone provide me with a description of such a construction, or a way to access such an article?
On the other hand, can anyone provide me with a proof, or a way to access a proof, that such a construction is impossible?
real-analysis
asked Mar 14 at 8:04
Michael FoxMichael Fox
13
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add a comment |
$begingroup$
A few years ago I saw a reference to two articles by two prominent 19th Century mathematicians, whose names I can no longer remember, to constructions of bicontinuous bijections between an interval, maybe open, maybe closed, of the reals, and a square, similarly open or closed, in 2-dim. Euclidean space, which I wouldn't have believed possible. However, I couldn't find any access to the articles. Can anyone provide me with a description of such a construction, or a way to access such an article?
On the other hand, can anyone provide me with a proof, or a way to access a proof, that such a construction is impossible?
real-analysis
asked Mar 14 at 8:04
Michael FoxMichael Fox
13
New contributor
Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Do you mean this?
$endgroup$
– poyea
Mar 14 at 8:07
add a comment |
$begingroup$
A few years ago I saw a reference to two articles by two prominent 19th Century mathematicians, whose names I can no longer remember, to constructions of bicontinuous bijections between an interval, maybe open, maybe closed, of the reals, and a square, similarly open or closed, in 2-dim. Euclidean space, which I wouldn't have believed possible. However, I couldn't find any access to the articles. Can anyone provide me with a description of such a construction, or a way to access such an article?
On the other hand, can anyone provide me with a proof, or a way to access a proof, that such a construction is impossible?
real-analysis
asked Mar 14 at 8:04
Michael FoxMichael Fox
13
New contributor
Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
A few years ago I saw a reference to two articles by two prominent 19th Century mathematicians, whose names I can no longer remember, to constructions of bicontinuous bijections between an interval, maybe open, maybe closed, of the reals, and a square, similarly open or closed, in 2-dim. Euclidean space, which I wouldn't have believed possible. However, I couldn't find any access to the articles. Can anyone provide me with a description of such a construction, or a way to access such an article?
On the other hand, can anyone provide me with a proof, or a way to access a proof, that such a construction is impossible?
real-analysis
real-analysis
asked Mar 14 at 8:04
Michael FoxMichael Fox
13
New contributor
Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 14 at 8:04
Michael FoxMichael Fox
13
New contributor
Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 14 at 8:04
Michael FoxMichael Fox
13
New contributor
Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 14 at 8:04
Michael FoxMichael Fox
13
asked Mar 14 at 8:04
Michael FoxMichael Fox
13
13
New contributor
Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Michael Fox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
Do you mean this?
$endgroup$
– poyea
Mar 14 at 8:07
add a comment |
$begingroup$
Do you mean this?
$endgroup$
– poyea
Mar 14 at 8:07
$begingroup$
Do you mean this?
$endgroup$
– poyea
Mar 14 at 8:07
$begingroup$
Do you mean this?
$endgroup$
– poyea
Mar 14 at 8:07
add a comment |
1 Answer
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No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.
answered Mar 14 at 8:07
Kavi Rama MurthyKavi Rama Murthy
68.7k53169
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add a comment |
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No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.
answered Mar 14 at 8:07
Kavi Rama MurthyKavi Rama Murthy
68.7k53169
$endgroup$
add a comment |
$begingroup$
No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.
answered Mar 14 at 8:07
Kavi Rama MurthyKavi Rama Murthy
68.7k53169
$endgroup$
add a comment |
$begingroup$
No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.
answered Mar 14 at 8:07
Kavi Rama MurthyKavi Rama Murthy
68.7k53169
$endgroup$
No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.
answered Mar 14 at 8:07
Kavi Rama MurthyKavi Rama Murthy
68.7k53169
answered Mar 14 at 8:07
Kavi Rama MurthyKavi Rama Murthy
68.7k53169
answered Mar 14 at 8:07
Kavi Rama MurthyKavi Rama Murthy
68.7k53169
answered Mar 14 at 8:07
Kavi Rama MurthyKavi Rama Murthy
68.7k53169
68.7k53169
add a comment |
add a comment |
Michael Fox is a new contributor. Be nice, and check out our Code of Conduct.
Michael Fox is a new contributor. Be nice, and check out our Code of Conduct.
Michael Fox is a new contributor. Be nice, and check out our Code of Conduct.
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Do you mean this?
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– poyea
Mar 14 at 8:07