Theorems in the form of “if and only if” such that the proof of one direction is extremely EASY to prove...
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I believe this is a common phenomenon in mathematics, but surprisingly, no such list has been created on this site. I don't know if it's of value, just out of curiosity, I want to see more examples. So my request is:
Theorems in the form of "if and only if" that the proof of one direction is extremely EASY, or intuitive, or make use of some standard techniques (e.g. diagram chasing), while the other one is extremely HARD, or counterintuitive, or require a certain amount of creativity. The 'if and only if' formulation should be as natural as possible.
Thanks in advance.
Edit: I know this question is somewhat ill-posed. A better one: Theorems in the form of “if and only if” such that the proofs of BOTH directions are nontrivial
soft-question big-list
asked Jan 11 at 5:07
YuiTo ChengYuiTo Cheng
279115
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put on hold as too broad by BlueRaja - Danny Pflughoeft, Arnaud D., Cesareo, Mees de Vries, Carl Mummert 2 days ago
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
|
show 3 more comments
$begingroup$
I believe this is a common phenomenon in mathematics, but surprisingly, no such list has been created on this site. I don't know if it's of value, just out of curiosity, I want to see more examples. So my request is:
Theorems in the form of "if and only if" that the proof of one direction is extremely EASY, or intuitive, or make use of some standard techniques (e.g. diagram chasing), while the other one is extremely HARD, or counterintuitive, or require a certain amount of creativity. The 'if and only if' formulation should be as natural as possible.
Thanks in advance.
Edit: I know this question is somewhat ill-posed. A better one: Theorems in the form of “if and only if” such that the proofs of BOTH directions are nontrivial
soft-question big-list
asked Jan 11 at 5:07
YuiTo ChengYuiTo Cheng
279115
$endgroup$
put on hold as too broad by BlueRaja - Danny Pflughoeft, Arnaud D., Cesareo, Mees de Vries, Carl Mummert 2 days ago
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
3
$begingroup$
One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
$endgroup$
– Blue
2 days ago
2
$begingroup$
That's a good metaphor!
$endgroup$
– YuiTo Cheng
2 days ago
11
$begingroup$
The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
$endgroup$
– bof
2 days ago
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@bof done, see math.stackexchange.com/questions/3069590/…
$endgroup$
– YuiTo Cheng
2 days ago
3
$begingroup$
You cannot make such a question more specific. You are asking for a list of things of a certain nature, where that set is known to be enormous. Any answer is either just one among many with no way to determine a clear "best" or requires thousands of pages to sufficiently answer in one go. This is textbook classic VTC:TB.
$endgroup$
– Nij
2 days ago
|
show 3 more comments
$begingroup$
I believe this is a common phenomenon in mathematics, but surprisingly, no such list has been created on this site. I don't know if it's of value, just out of curiosity, I want to see more examples. So my request is:
Theorems in the form of "if and only if" that the proof of one direction is extremely EASY, or intuitive, or make use of some standard techniques (e.g. diagram chasing), while the other one is extremely HARD, or counterintuitive, or require a certain amount of creativity. The 'if and only if' formulation should be as natural as possible.
Thanks in advance.
Edit: I know this question is somewhat ill-posed. A better one: Theorems in the form of “if and only if” such that the proofs of BOTH directions are nontrivial
soft-question big-list
asked Jan 11 at 5:07
YuiTo ChengYuiTo Cheng
279115
$endgroup$
I believe this is a common phenomenon in mathematics, but surprisingly, no such list has been created on this site. I don't know if it's of value, just out of curiosity, I want to see more examples. So my request is:
Theorems in the form of "if and only if" that the proof of one direction is extremely EASY, or intuitive, or make use of some standard techniques (e.g. diagram chasing), while the other one is extremely HARD, or counterintuitive, or require a certain amount of creativity. The 'if and only if' formulation should be as natural as possible.
Thanks in advance.
Edit: I know this question is somewhat ill-posed. A better one: Theorems in the form of “if and only if” such that the proofs of BOTH directions are nontrivial
soft-question big-list
soft-question big-list
asked Jan 11 at 5:07
YuiTo ChengYuiTo Cheng
279115
asked Jan 11 at 5:07
YuiTo ChengYuiTo Cheng
279115
edited 2 days ago
YuiTo Cheng
asked Jan 11 at 5:07
YuiTo ChengYuiTo Cheng
279115
asked Jan 11 at 5:07
YuiTo ChengYuiTo Cheng
279115
asked Jan 11 at 5:07
YuiTo ChengYuiTo Cheng
279115
279115
put on hold as too broad by BlueRaja - Danny Pflughoeft, Arnaud D., Cesareo, Mees de Vries, Carl Mummert 2 days ago
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as too broad by BlueRaja - Danny Pflughoeft, Arnaud D., Cesareo, Mees de Vries, Carl Mummert 2 days ago
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
3
$begingroup$
One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
$endgroup$
– Blue
2 days ago
2
$begingroup$
That's a good metaphor!
$endgroup$
– YuiTo Cheng
2 days ago
11
$begingroup$
The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
$endgroup$
– bof
2 days ago
$begingroup$
@bof done, see math.stackexchange.com/questions/3069590/…
$endgroup$
– YuiTo Cheng
2 days ago
3
$begingroup$
You cannot make such a question more specific. You are asking for a list of things of a certain nature, where that set is known to be enormous. Any answer is either just one among many with no way to determine a clear "best" or requires thousands of pages to sufficiently answer in one go. This is textbook classic VTC:TB.
$endgroup$
– Nij
2 days ago
|
show 3 more comments
3
$begingroup$
One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
$endgroup$
– Blue
2 days ago
2
$begingroup$
That's a good metaphor!
$endgroup$
– YuiTo Cheng
2 days ago
11
$begingroup$
The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
$endgroup$
– bof
2 days ago
$begingroup$
@bof done, see math.stackexchange.com/questions/3069590/…
$endgroup$
– YuiTo Cheng
2 days ago
3
$begingroup$
You cannot make such a question more specific. You are asking for a list of things of a certain nature, where that set is known to be enormous. Any answer is either just one among many with no way to determine a clear "best" or requires thousands of pages to sufficiently answer in one go. This is textbook classic VTC:TB.
$endgroup$
– Nij
2 days ago
3
3
$begingroup$
One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
$endgroup$
– Blue
2 days ago
$begingroup$
One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
$endgroup$
– Blue
2 days ago
2
2
$begingroup$
That's a good metaphor!
$endgroup$
– YuiTo Cheng
2 days ago
$begingroup$
That's a good metaphor!
$endgroup$
– YuiTo Cheng
2 days ago
11
11
$begingroup$
The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
$endgroup$
– bof
2 days ago
$begingroup$
The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
$endgroup$
– bof
2 days ago
$begingroup$
@bof done, see math.stackexchange.com/questions/3069590/…
$endgroup$
– YuiTo Cheng
2 days ago
$begingroup$
@bof done, see math.stackexchange.com/questions/3069590/…
$endgroup$
– YuiTo Cheng
2 days ago
3
3
$begingroup$
You cannot make such a question more specific. You are asking for a list of things of a certain nature, where that set is known to be enormous. Any answer is either just one among many with no way to determine a clear "best" or requires thousands of pages to sufficiently answer in one go. This is textbook classic VTC:TB.
$endgroup$
– Nij
2 days ago
$begingroup$
You cannot make such a question more specific. You are asking for a list of things of a certain nature, where that set is known to be enormous. Any answer is either just one among many with no way to determine a clear "best" or requires thousands of pages to sufficiently answer in one go. This is textbook classic VTC:TB.
$endgroup$
– Nij
2 days ago
|
show 3 more comments
9 Answers
9
active
oldest
votes
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"A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere."
One direction [only if] is the Poincaré conjecture. The other direction shouldn't be too bad hopefully.
answered 2 days ago
Alvin JinAlvin Jin
2,2121019
$endgroup$
add a comment |
$begingroup$
Let $n$ be a positive integer. The equation $x^n+y^n=z^n$ is solvable in positive integers $x,y,z$ iff $nle2$.
answered 2 days ago
bofbof
50.8k457120
$endgroup$
add a comment |
$begingroup$
The bisectors of two angles of a triangle are of equal length if and only if the two bisected angles are equal. If the two angles are equal, the triangle is isosceles and the proof is very easy. However proving that a triangle must be isosceles if the bisectors of two of its angles are of equal length seems to be quite difficult.
answered 2 days ago
lonza leggieralonza leggiera
3484
New contributor
lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
The Steiner–Lehmus theorem.
$endgroup$
– Rosie F
2 days ago
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This one is deeper than I thought, especially the nonexistence of a "direct proof".
$endgroup$
– YuiTo Cheng
2 days ago
add a comment |
$begingroup$
All planar graphs are $n$-colorable iff $nge4$.
answered 2 days ago
bofbof
50.8k457120
$endgroup$
add a comment |
$begingroup$
Integers $a * b = 944871836856449473$ and $b > a > 1$
iff
$a = 961748941$ and $b = 982451653$
If is trivial multiplication. Only if requires large prime factorization.
answered 2 days ago
MooseBoysMooseBoys
1895
$endgroup$
10
$begingroup$
The hard direction only requires primality testing, not factorization.
$endgroup$
– bof
2 days ago
add a comment |
$begingroup$
The difficult part is not as difficult as some of the other answers, but perhaps the "if and only if" formulation is more natural: Kuratowski's theorem that a graph is planar if and only if it does not have a subgraph which is a subdivision of either $K_{3,3}$ or $K_5$.
answered 2 days ago
Especially LimeEspecially Lime
21.8k22858
$endgroup$
add a comment |
$begingroup$
The product of topological spaces is compact if and only if all of them are compact. One direction is trivial because the projections are continuous and surjective functions. The another direction, well, is the axiom of choice.
answered 2 days ago
Carlos JiménezCarlos Jiménez
2,3851520
$endgroup$
2
$begingroup$
Assuming they're all non-empty.
$endgroup$
– Arnaud D.
2 days ago
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Actually, the projections always being surjective is also equivalent to the axiom of choice.
$endgroup$
– Marc Paul
2 days ago
add a comment |
$begingroup$
An even integer $n$ is the sum of two primes iff $n>2$.
("If" is Goldbach's conjecture, which is still open. "Only if" is trivial.)
answered 2 days ago
Rosie FRosie F
1,262314
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5
$begingroup$
There may be chances that Goldbach's conjecture is wrong...
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– YuiTo Cheng
2 days ago
add a comment |
$begingroup$
$mathbb R^n$ has the structure of a real division algebra iff $n=1,2,4$ or $8$.
answered 2 days ago
guest9366710guest9366710
111
New contributor
guest9366710 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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add a comment |
9 Answers
9
active
oldest
votes
9 Answers
9
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
"A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere."
One direction [only if] is the Poincaré conjecture. The other direction shouldn't be too bad hopefully.
answered 2 days ago
Alvin JinAlvin Jin
2,2121019
$endgroup$
add a comment |
$begingroup$
"A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere."
One direction [only if] is the Poincaré conjecture. The other direction shouldn't be too bad hopefully.
answered 2 days ago
Alvin JinAlvin Jin
2,2121019
$endgroup$
add a comment |
$begingroup$
"A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere."
One direction [only if] is the Poincaré conjecture. The other direction shouldn't be too bad hopefully.
answered 2 days ago
Alvin JinAlvin Jin
2,2121019
$endgroup$
"A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere."
One direction [only if] is the Poincaré conjecture. The other direction shouldn't be too bad hopefully.
answered 2 days ago
Alvin JinAlvin Jin
2,2121019
answered 2 days ago
Alvin JinAlvin Jin
2,2121019
answered 2 days ago
Alvin JinAlvin Jin
2,2121019
answered 2 days ago
Alvin JinAlvin Jin
2,2121019
2,2121019
add a comment |
add a comment |
$begingroup$
Let $n$ be a positive integer. The equation $x^n+y^n=z^n$ is solvable in positive integers $x,y,z$ iff $nle2$.
answered 2 days ago
bofbof
50.8k457120
$endgroup$
add a comment |
$begingroup$
Let $n$ be a positive integer. The equation $x^n+y^n=z^n$ is solvable in positive integers $x,y,z$ iff $nle2$.
answered 2 days ago
bofbof
50.8k457120
$endgroup$
add a comment |
$begingroup$
Let $n$ be a positive integer. The equation $x^n+y^n=z^n$ is solvable in positive integers $x,y,z$ iff $nle2$.
answered 2 days ago
bofbof
50.8k457120
$endgroup$
Let $n$ be a positive integer. The equation $x^n+y^n=z^n$ is solvable in positive integers $x,y,z$ iff $nle2$.
answered 2 days ago
bofbof
50.8k457120
answered 2 days ago
bofbof
50.8k457120
answered 2 days ago
bofbof
50.8k457120
answered 2 days ago
bofbof
50.8k457120
50.8k457120
add a comment |
add a comment |
$begingroup$
The bisectors of two angles of a triangle are of equal length if and only if the two bisected angles are equal. If the two angles are equal, the triangle is isosceles and the proof is very easy. However proving that a triangle must be isosceles if the bisectors of two of its angles are of equal length seems to be quite difficult.
answered 2 days ago
lonza leggieralonza leggiera
3484
New contributor
lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
The Steiner–Lehmus theorem.
$endgroup$
– Rosie F
2 days ago
$begingroup$
This one is deeper than I thought, especially the nonexistence of a "direct proof".
$endgroup$
– YuiTo Cheng
2 days ago
add a comment |
$begingroup$
The bisectors of two angles of a triangle are of equal length if and only if the two bisected angles are equal. If the two angles are equal, the triangle is isosceles and the proof is very easy. However proving that a triangle must be isosceles if the bisectors of two of its angles are of equal length seems to be quite difficult.
answered 2 days ago
lonza leggieralonza leggiera
3484
New contributor
lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
The Steiner–Lehmus theorem.
$endgroup$
– Rosie F
2 days ago
$begingroup$
This one is deeper than I thought, especially the nonexistence of a "direct proof".
$endgroup$
– YuiTo Cheng
2 days ago
add a comment |
$begingroup$
The bisectors of two angles of a triangle are of equal length if and only if the two bisected angles are equal. If the two angles are equal, the triangle is isosceles and the proof is very easy. However proving that a triangle must be isosceles if the bisectors of two of its angles are of equal length seems to be quite difficult.
answered 2 days ago
lonza leggieralonza leggiera
3484
New contributor
lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
The bisectors of two angles of a triangle are of equal length if and only if the two bisected angles are equal. If the two angles are equal, the triangle is isosceles and the proof is very easy. However proving that a triangle must be isosceles if the bisectors of two of its angles are of equal length seems to be quite difficult.
answered 2 days ago
lonza leggieralonza leggiera
3484
New contributor
lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 days ago
lonza leggieralonza leggiera
3484
New contributor
lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 days ago
lonza leggieralonza leggiera
3484
answered 2 days ago
lonza leggieralonza leggiera
3484
3484
New contributor
lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
lonza leggiera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
$begingroup$
The Steiner–Lehmus theorem.
$endgroup$
– Rosie F
2 days ago
$begingroup$
This one is deeper than I thought, especially the nonexistence of a "direct proof".
$endgroup$
– YuiTo Cheng
2 days ago
add a comment |
2
$begingroup$
The Steiner–Lehmus theorem.
$endgroup$
– Rosie F
2 days ago
$begingroup$
This one is deeper than I thought, especially the nonexistence of a "direct proof".
$endgroup$
– YuiTo Cheng
2 days ago
2
2
$begingroup$
The Steiner–Lehmus theorem.
$endgroup$
– Rosie F
2 days ago
$begingroup$
The Steiner–Lehmus theorem.
$endgroup$
– Rosie F
2 days ago
$begingroup$
This one is deeper than I thought, especially the nonexistence of a "direct proof".
$endgroup$
– YuiTo Cheng
2 days ago
$begingroup$
This one is deeper than I thought, especially the nonexistence of a "direct proof".
$endgroup$
– YuiTo Cheng
2 days ago
add a comment |
$begingroup$
All planar graphs are $n$-colorable iff $nge4$.
answered 2 days ago
bofbof
50.8k457120
$endgroup$
add a comment |
$begingroup$
All planar graphs are $n$-colorable iff $nge4$.
answered 2 days ago
bofbof
50.8k457120
$endgroup$
add a comment |
$begingroup$
All planar graphs are $n$-colorable iff $nge4$.
answered 2 days ago
bofbof
50.8k457120
$endgroup$
All planar graphs are $n$-colorable iff $nge4$.
answered 2 days ago
bofbof
50.8k457120
answered 2 days ago
bofbof
50.8k457120
answered 2 days ago
bofbof
50.8k457120
answered 2 days ago
bofbof
50.8k457120
50.8k457120
add a comment |
add a comment |
$begingroup$
Integers $a * b = 944871836856449473$ and $b > a > 1$
iff
$a = 961748941$ and $b = 982451653$
If is trivial multiplication. Only if requires large prime factorization.
answered 2 days ago
MooseBoysMooseBoys
1895
$endgroup$
10
$begingroup$
The hard direction only requires primality testing, not factorization.
$endgroup$
– bof
2 days ago
add a comment |
$begingroup$
Integers $a * b = 944871836856449473$ and $b > a > 1$
iff
$a = 961748941$ and $b = 982451653$
If is trivial multiplication. Only if requires large prime factorization.
answered 2 days ago
MooseBoysMooseBoys
1895
$endgroup$
10
$begingroup$
The hard direction only requires primality testing, not factorization.
$endgroup$
– bof
2 days ago
add a comment |
$begingroup$
Integers $a * b = 944871836856449473$ and $b > a > 1$
iff
$a = 961748941$ and $b = 982451653$
If is trivial multiplication. Only if requires large prime factorization.
answered 2 days ago
MooseBoysMooseBoys
1895
$endgroup$
Integers $a * b = 944871836856449473$ and $b > a > 1$
iff
$a = 961748941$ and $b = 982451653$
If is trivial multiplication. Only if requires large prime factorization.
answered 2 days ago
MooseBoysMooseBoys
1895
edited 2 days ago
answered 2 days ago
MooseBoysMooseBoys
1895
answered 2 days ago
MooseBoysMooseBoys
1895
answered 2 days ago
MooseBoysMooseBoys
1895
1895
10
$begingroup$
The hard direction only requires primality testing, not factorization.
$endgroup$
– bof
2 days ago
add a comment |
10
$begingroup$
The hard direction only requires primality testing, not factorization.
$endgroup$
– bof
2 days ago
10
10
$begingroup$
The hard direction only requires primality testing, not factorization.
$endgroup$
– bof
2 days ago
$begingroup$
The hard direction only requires primality testing, not factorization.
$endgroup$
– bof
2 days ago
add a comment |
$begingroup$
The difficult part is not as difficult as some of the other answers, but perhaps the "if and only if" formulation is more natural: Kuratowski's theorem that a graph is planar if and only if it does not have a subgraph which is a subdivision of either $K_{3,3}$ or $K_5$.
answered 2 days ago
Especially LimeEspecially Lime
21.8k22858
$endgroup$
add a comment |
$begingroup$
The difficult part is not as difficult as some of the other answers, but perhaps the "if and only if" formulation is more natural: Kuratowski's theorem that a graph is planar if and only if it does not have a subgraph which is a subdivision of either $K_{3,3}$ or $K_5$.
answered 2 days ago
Especially LimeEspecially Lime
21.8k22858
$endgroup$
add a comment |
$begingroup$
The difficult part is not as difficult as some of the other answers, but perhaps the "if and only if" formulation is more natural: Kuratowski's theorem that a graph is planar if and only if it does not have a subgraph which is a subdivision of either $K_{3,3}$ or $K_5$.
answered 2 days ago
Especially LimeEspecially Lime
21.8k22858
$endgroup$
The difficult part is not as difficult as some of the other answers, but perhaps the "if and only if" formulation is more natural: Kuratowski's theorem that a graph is planar if and only if it does not have a subgraph which is a subdivision of either $K_{3,3}$ or $K_5$.
answered 2 days ago
Especially LimeEspecially Lime
21.8k22858
edited 2 days ago
answered 2 days ago
Especially LimeEspecially Lime
21.8k22858
answered 2 days ago
Especially LimeEspecially Lime
21.8k22858
answered 2 days ago
Especially LimeEspecially Lime
21.8k22858
21.8k22858
add a comment |
add a comment |
$begingroup$
The product of topological spaces is compact if and only if all of them are compact. One direction is trivial because the projections are continuous and surjective functions. The another direction, well, is the axiom of choice.
answered 2 days ago
Carlos JiménezCarlos Jiménez
2,3851520
$endgroup$
2
$begingroup$
Assuming they're all non-empty.
$endgroup$
– Arnaud D.
2 days ago
$begingroup$
Actually, the projections always being surjective is also equivalent to the axiom of choice.
$endgroup$
– Marc Paul
2 days ago
add a comment |
$begingroup$
The product of topological spaces is compact if and only if all of them are compact. One direction is trivial because the projections are continuous and surjective functions. The another direction, well, is the axiom of choice.
answered 2 days ago
Carlos JiménezCarlos Jiménez
2,3851520
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2
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Assuming they're all non-empty.
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– Arnaud D.
2 days ago
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Actually, the projections always being surjective is also equivalent to the axiom of choice.
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– Marc Paul
2 days ago
add a comment |
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The product of topological spaces is compact if and only if all of them are compact. One direction is trivial because the projections are continuous and surjective functions. The another direction, well, is the axiom of choice.
answered 2 days ago
Carlos JiménezCarlos Jiménez
2,3851520
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The product of topological spaces is compact if and only if all of them are compact. One direction is trivial because the projections are continuous and surjective functions. The another direction, well, is the axiom of choice.
answered 2 days ago
Carlos JiménezCarlos Jiménez
2,3851520
answered 2 days ago
Carlos JiménezCarlos Jiménez
2,3851520
answered 2 days ago
Carlos JiménezCarlos Jiménez
2,3851520
answered 2 days ago
Carlos JiménezCarlos Jiménez
2,3851520
2,3851520
2
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Assuming they're all non-empty.
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– Arnaud D.
2 days ago
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Actually, the projections always being surjective is also equivalent to the axiom of choice.
$endgroup$
– Marc Paul
2 days ago
add a comment |
2
$begingroup$
Assuming they're all non-empty.
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– Arnaud D.
2 days ago
$begingroup$
Actually, the projections always being surjective is also equivalent to the axiom of choice.
$endgroup$
– Marc Paul
2 days ago
2
2
$begingroup$
Assuming they're all non-empty.
$endgroup$
– Arnaud D.
2 days ago
$begingroup$
Assuming they're all non-empty.
$endgroup$
– Arnaud D.
2 days ago
$begingroup$
Actually, the projections always being surjective is also equivalent to the axiom of choice.
$endgroup$
– Marc Paul
2 days ago
$begingroup$
Actually, the projections always being surjective is also equivalent to the axiom of choice.
$endgroup$
– Marc Paul
2 days ago
add a comment |
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An even integer $n$ is the sum of two primes iff $n>2$.
("If" is Goldbach's conjecture, which is still open. "Only if" is trivial.)
answered 2 days ago
Rosie FRosie F
1,262314
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5
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There may be chances that Goldbach's conjecture is wrong...
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– YuiTo Cheng
2 days ago
add a comment |
$begingroup$
An even integer $n$ is the sum of two primes iff $n>2$.
("If" is Goldbach's conjecture, which is still open. "Only if" is trivial.)
answered 2 days ago
Rosie FRosie F
1,262314
$endgroup$
5
$begingroup$
There may be chances that Goldbach's conjecture is wrong...
$endgroup$
– YuiTo Cheng
2 days ago
add a comment |
$begingroup$
An even integer $n$ is the sum of two primes iff $n>2$.
("If" is Goldbach's conjecture, which is still open. "Only if" is trivial.)
answered 2 days ago
Rosie FRosie F
1,262314
$endgroup$
An even integer $n$ is the sum of two primes iff $n>2$.
("If" is Goldbach's conjecture, which is still open. "Only if" is trivial.)
answered 2 days ago
Rosie FRosie F
1,262314
answered 2 days ago
Rosie FRosie F
1,262314
answered 2 days ago
Rosie FRosie F
1,262314
answered 2 days ago
Rosie FRosie F
1,262314
1,262314
5
$begingroup$
There may be chances that Goldbach's conjecture is wrong...
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– YuiTo Cheng
2 days ago
add a comment |
5
$begingroup$
There may be chances that Goldbach's conjecture is wrong...
$endgroup$
– YuiTo Cheng
2 days ago
5
5
$begingroup$
There may be chances that Goldbach's conjecture is wrong...
$endgroup$
– YuiTo Cheng
2 days ago
$begingroup$
There may be chances that Goldbach's conjecture is wrong...
$endgroup$
– YuiTo Cheng
2 days ago
add a comment |
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$mathbb R^n$ has the structure of a real division algebra iff $n=1,2,4$ or $8$.
answered 2 days ago
guest9366710guest9366710
111
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add a comment |
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$mathbb R^n$ has the structure of a real division algebra iff $n=1,2,4$ or $8$.
answered 2 days ago
guest9366710guest9366710
111
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guest9366710 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
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$mathbb R^n$ has the structure of a real division algebra iff $n=1,2,4$ or $8$.
answered 2 days ago
guest9366710guest9366710
111
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guest9366710 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$mathbb R^n$ has the structure of a real division algebra iff $n=1,2,4$ or $8$.
answered 2 days ago
guest9366710guest9366710
111
New contributor
guest9366710 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 days ago
guest9366710guest9366710
111
New contributor
guest9366710 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 2 days ago
guest9366710guest9366710
111
answered 2 days ago
guest9366710guest9366710
111
111
New contributor
guest9366710 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
guest9366710 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
guest9366710 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
3
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One of my college professors likened this phenomenon to running one's palm along one's face. Down is easy; up is not.
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– Blue
2 days ago
2
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That's a good metaphor!
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– YuiTo Cheng
2 days ago
11
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The reason nobody has created such a list may be the fact that most (nontrivial) "iff" theorems belong on your list. It would be more interesting to see a list of "iff" theorems where both directions are nontrivial.
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– bof
2 days ago
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@bof done, see math.stackexchange.com/questions/3069590/…
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– YuiTo Cheng
2 days ago
3
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You cannot make such a question more specific. You are asking for a list of things of a certain nature, where that set is known to be enormous. Any answer is either just one among many with no way to determine a clear "best" or requires thousands of pages to sufficiently answer in one go. This is textbook classic VTC:TB.
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– Nij
2 days ago