How to be good at coming up with counter example in Topology
$begingroup$
This is a more generalized question, but does anyone have a set of tips or tricks to come up with distinctive examples and counterexamples in Topology and Analysis? More specific, how can people often come up with exotic sequence or mappings between spaces? I can understand the intuition behind some of the simple fractions in the by playing with simple fractions like $frac{1}{n}$, but it seems bizarre to me at this moment how people just come up with maps involving complex numbers, trigonometry between exotic spaces out of nowhere
general-topology examples-counterexamples intuition
asked Apr 25 at 22:18
Joe MartinJoe Martin
1287
$endgroup$
add a comment |
$begingroup$
This is a more generalized question, but does anyone have a set of tips or tricks to come up with distinctive examples and counterexamples in Topology and Analysis? More specific, how can people often come up with exotic sequence or mappings between spaces? I can understand the intuition behind some of the simple fractions in the by playing with simple fractions like $frac{1}{n}$, but it seems bizarre to me at this moment how people just come up with maps involving complex numbers, trigonometry between exotic spaces out of nowhere
general-topology examples-counterexamples intuition
asked Apr 25 at 22:18
Joe MartinJoe Martin
1287
$endgroup$
$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
Apr 25 at 22:43
$begingroup$
Another general question: How to attack “if true, prove it; if not true, give a counterexample” question?
$endgroup$
– YuiTo Cheng
Apr 26 at 3:58
$begingroup$
It can be difficult. When, in the latter 19th century, Weierstrass exhibited a continuous nowhere-differentiable $f:Bbb R to Bbb R$, many were surprised as many expected that to be impossible.
$endgroup$
– DanielWainfleet
Apr 26 at 11:24
add a comment |
$begingroup$
This is a more generalized question, but does anyone have a set of tips or tricks to come up with distinctive examples and counterexamples in Topology and Analysis? More specific, how can people often come up with exotic sequence or mappings between spaces? I can understand the intuition behind some of the simple fractions in the by playing with simple fractions like $frac{1}{n}$, but it seems bizarre to me at this moment how people just come up with maps involving complex numbers, trigonometry between exotic spaces out of nowhere
general-topology examples-counterexamples intuition
asked Apr 25 at 22:18
Joe MartinJoe Martin
1287
$endgroup$
This is a more generalized question, but does anyone have a set of tips or tricks to come up with distinctive examples and counterexamples in Topology and Analysis? More specific, how can people often come up with exotic sequence or mappings between spaces? I can understand the intuition behind some of the simple fractions in the by playing with simple fractions like $frac{1}{n}$, but it seems bizarre to me at this moment how people just come up with maps involving complex numbers, trigonometry between exotic spaces out of nowhere
general-topology examples-counterexamples intuition
general-topology examples-counterexamples intuition
asked Apr 25 at 22:18
Joe MartinJoe Martin
1287
asked Apr 25 at 22:18
Joe MartinJoe Martin
1287
asked Apr 25 at 22:18
Joe MartinJoe Martin
1287
asked Apr 25 at 22:18
Joe MartinJoe Martin
1287
asked Apr 25 at 22:18
Joe MartinJoe Martin
1287
1287
$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
Apr 25 at 22:43
$begingroup$
Another general question: How to attack “if true, prove it; if not true, give a counterexample” question?
$endgroup$
– YuiTo Cheng
Apr 26 at 3:58
$begingroup$
It can be difficult. When, in the latter 19th century, Weierstrass exhibited a continuous nowhere-differentiable $f:Bbb R to Bbb R$, many were surprised as many expected that to be impossible.
$endgroup$
– DanielWainfleet
Apr 26 at 11:24
add a comment |
$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
Apr 25 at 22:43
$begingroup$
Another general question: How to attack “if true, prove it; if not true, give a counterexample” question?
$endgroup$
– YuiTo Cheng
Apr 26 at 3:58
$begingroup$
It can be difficult. When, in the latter 19th century, Weierstrass exhibited a continuous nowhere-differentiable $f:Bbb R to Bbb R$, many were surprised as many expected that to be impossible.
$endgroup$
– DanielWainfleet
Apr 26 at 11:24
$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
Apr 25 at 22:43
$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
Apr 25 at 22:43
$begingroup$
Another general question: How to attack “if true, prove it; if not true, give a counterexample” question?
$endgroup$
– YuiTo Cheng
Apr 26 at 3:58
$begingroup$
Another general question: How to attack “if true, prove it; if not true, give a counterexample” question?
$endgroup$
– YuiTo Cheng
Apr 26 at 3:58
$begingroup$
It can be difficult. When, in the latter 19th century, Weierstrass exhibited a continuous nowhere-differentiable $f:Bbb R to Bbb R$, many were surprised as many expected that to be impossible.
$endgroup$
– DanielWainfleet
Apr 26 at 11:24
$begingroup$
It can be difficult. When, in the latter 19th century, Weierstrass exhibited a continuous nowhere-differentiable $f:Bbb R to Bbb R$, many were surprised as many expected that to be impossible.
$endgroup$
– DanielWainfleet
Apr 26 at 11:24
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
For counterexamples, just think: "mission sabotage". In other words, deliberately try to break a given statement.
There are generally no "tips", "tricks", "recipes", or anything else of a universal caliber. (When there are, they are so valued that you will surely run across them.) Mathematics is an art as much as it is a science: one tries, examines for errors, and corrects if needed, as many times as it takes.
The best there is in the direction you are asking is learning a sufficiently rich arsenal of counterexamples. To help with that, Olmsted and Gelbaum have written Counterexamples in Analysis, which is a great and highly beneficial read.
answered Apr 25 at 22:24
avsavs
5,0061515
$endgroup$
3
$begingroup$
There is similarly a book titled Counterexamples in Topology. Also, it might help to think about what properties you are implicitly assuming when you try to come up with examples. E.g., Am I only looking at continuous functions? Differentiable functions? Increasing functions? Compact spaces? Subsets of $mathbb{R}^n$? Metric spaces? Hausdorff spaces?
$endgroup$
– kccu
Apr 25 at 22:27
1
$begingroup$
Thank you very much for the comment. This seems like an excellent book for me to start with. It is just I started taking my first algebraic topology course this semester, but I feel dull as every time I think about some possible theorem to prove, the stack-exchange community would just come up with bizarre (at least to me) counter-example in a short period that would take forever for me to verify.
$endgroup$
– Joe Martin
Apr 25 at 23:35
$begingroup$
@Joe Martin, that's because they say it (or a similar counterexample) before. Don't judge your ability to conceive of brand new ideas against the experience of the entire MSE community.
$endgroup$
– Mark S.
Apr 26 at 11:20
add a comment |
$begingroup$
I think that it would indeed be odd for people to come up with exotic counterexamples to innocuous conjectures out of nowhere, as you say. Really, what is guiding those counterexamples is a lot of time and experience spent with problems and the material. When you are reading the statement of a theorem, try seeing what happens when you omit a hypothesis to see what may go wrong, and talk to people about it, either here online or in person to share your thoughts. The more you learn, the more connections you will make, and eventually you will begin to see more as you synthesize that knowledge.
answered Apr 25 at 22:27
Alex OrtizAlex Ortiz
11.8k21544
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
For counterexamples, just think: "mission sabotage". In other words, deliberately try to break a given statement.
There are generally no "tips", "tricks", "recipes", or anything else of a universal caliber. (When there are, they are so valued that you will surely run across them.) Mathematics is an art as much as it is a science: one tries, examines for errors, and corrects if needed, as many times as it takes.
The best there is in the direction you are asking is learning a sufficiently rich arsenal of counterexamples. To help with that, Olmsted and Gelbaum have written Counterexamples in Analysis, which is a great and highly beneficial read.
answered Apr 25 at 22:24
avsavs
5,0061515
$endgroup$
3
$begingroup$
There is similarly a book titled Counterexamples in Topology. Also, it might help to think about what properties you are implicitly assuming when you try to come up with examples. E.g., Am I only looking at continuous functions? Differentiable functions? Increasing functions? Compact spaces? Subsets of $mathbb{R}^n$? Metric spaces? Hausdorff spaces?
$endgroup$
– kccu
Apr 25 at 22:27
1
$begingroup$
Thank you very much for the comment. This seems like an excellent book for me to start with. It is just I started taking my first algebraic topology course this semester, but I feel dull as every time I think about some possible theorem to prove, the stack-exchange community would just come up with bizarre (at least to me) counter-example in a short period that would take forever for me to verify.
$endgroup$
– Joe Martin
Apr 25 at 23:35
$begingroup$
@Joe Martin, that's because they say it (or a similar counterexample) before. Don't judge your ability to conceive of brand new ideas against the experience of the entire MSE community.
$endgroup$
– Mark S.
Apr 26 at 11:20
add a comment |
$begingroup$
For counterexamples, just think: "mission sabotage". In other words, deliberately try to break a given statement.
There are generally no "tips", "tricks", "recipes", or anything else of a universal caliber. (When there are, they are so valued that you will surely run across them.) Mathematics is an art as much as it is a science: one tries, examines for errors, and corrects if needed, as many times as it takes.
The best there is in the direction you are asking is learning a sufficiently rich arsenal of counterexamples. To help with that, Olmsted and Gelbaum have written Counterexamples in Analysis, which is a great and highly beneficial read.
answered Apr 25 at 22:24
avsavs
5,0061515
$endgroup$
3
$begingroup$
There is similarly a book titled Counterexamples in Topology. Also, it might help to think about what properties you are implicitly assuming when you try to come up with examples. E.g., Am I only looking at continuous functions? Differentiable functions? Increasing functions? Compact spaces? Subsets of $mathbb{R}^n$? Metric spaces? Hausdorff spaces?
$endgroup$
– kccu
Apr 25 at 22:27
1
$begingroup$
Thank you very much for the comment. This seems like an excellent book for me to start with. It is just I started taking my first algebraic topology course this semester, but I feel dull as every time I think about some possible theorem to prove, the stack-exchange community would just come up with bizarre (at least to me) counter-example in a short period that would take forever for me to verify.
$endgroup$
– Joe Martin
Apr 25 at 23:35
$begingroup$
@Joe Martin, that's because they say it (or a similar counterexample) before. Don't judge your ability to conceive of brand new ideas against the experience of the entire MSE community.
$endgroup$
– Mark S.
Apr 26 at 11:20
add a comment |
$begingroup$
For counterexamples, just think: "mission sabotage". In other words, deliberately try to break a given statement.
There are generally no "tips", "tricks", "recipes", or anything else of a universal caliber. (When there are, they are so valued that you will surely run across them.) Mathematics is an art as much as it is a science: one tries, examines for errors, and corrects if needed, as many times as it takes.
The best there is in the direction you are asking is learning a sufficiently rich arsenal of counterexamples. To help with that, Olmsted and Gelbaum have written Counterexamples in Analysis, which is a great and highly beneficial read.
answered Apr 25 at 22:24
avsavs
5,0061515
$endgroup$
For counterexamples, just think: "mission sabotage". In other words, deliberately try to break a given statement.
There are generally no "tips", "tricks", "recipes", or anything else of a universal caliber. (When there are, they are so valued that you will surely run across them.) Mathematics is an art as much as it is a science: one tries, examines for errors, and corrects if needed, as many times as it takes.
The best there is in the direction you are asking is learning a sufficiently rich arsenal of counterexamples. To help with that, Olmsted and Gelbaum have written Counterexamples in Analysis, which is a great and highly beneficial read.
answered Apr 25 at 22:24
avsavs
5,0061515
answered Apr 25 at 22:24
avsavs
5,0061515
answered Apr 25 at 22:24
avsavs
5,0061515
answered Apr 25 at 22:24
avsavs
5,0061515
5,0061515
3
$begingroup$
There is similarly a book titled Counterexamples in Topology. Also, it might help to think about what properties you are implicitly assuming when you try to come up with examples. E.g., Am I only looking at continuous functions? Differentiable functions? Increasing functions? Compact spaces? Subsets of $mathbb{R}^n$? Metric spaces? Hausdorff spaces?
$endgroup$
– kccu
Apr 25 at 22:27
1
$begingroup$
Thank you very much for the comment. This seems like an excellent book for me to start with. It is just I started taking my first algebraic topology course this semester, but I feel dull as every time I think about some possible theorem to prove, the stack-exchange community would just come up with bizarre (at least to me) counter-example in a short period that would take forever for me to verify.
$endgroup$
– Joe Martin
Apr 25 at 23:35
$begingroup$
@Joe Martin, that's because they say it (or a similar counterexample) before. Don't judge your ability to conceive of brand new ideas against the experience of the entire MSE community.
$endgroup$
– Mark S.
Apr 26 at 11:20
add a comment |
3
$begingroup$
There is similarly a book titled Counterexamples in Topology. Also, it might help to think about what properties you are implicitly assuming when you try to come up with examples. E.g., Am I only looking at continuous functions? Differentiable functions? Increasing functions? Compact spaces? Subsets of $mathbb{R}^n$? Metric spaces? Hausdorff spaces?
$endgroup$
– kccu
Apr 25 at 22:27
1
$begingroup$
Thank you very much for the comment. This seems like an excellent book for me to start with. It is just I started taking my first algebraic topology course this semester, but I feel dull as every time I think about some possible theorem to prove, the stack-exchange community would just come up with bizarre (at least to me) counter-example in a short period that would take forever for me to verify.
$endgroup$
– Joe Martin
Apr 25 at 23:35
$begingroup$
@Joe Martin, that's because they say it (or a similar counterexample) before. Don't judge your ability to conceive of brand new ideas against the experience of the entire MSE community.
$endgroup$
– Mark S.
Apr 26 at 11:20
3
3
$begingroup$
There is similarly a book titled Counterexamples in Topology. Also, it might help to think about what properties you are implicitly assuming when you try to come up with examples. E.g., Am I only looking at continuous functions? Differentiable functions? Increasing functions? Compact spaces? Subsets of $mathbb{R}^n$? Metric spaces? Hausdorff spaces?
$endgroup$
– kccu
Apr 25 at 22:27
$begingroup$
There is similarly a book titled Counterexamples in Topology. Also, it might help to think about what properties you are implicitly assuming when you try to come up with examples. E.g., Am I only looking at continuous functions? Differentiable functions? Increasing functions? Compact spaces? Subsets of $mathbb{R}^n$? Metric spaces? Hausdorff spaces?
$endgroup$
– kccu
Apr 25 at 22:27
1
1
$begingroup$
Thank you very much for the comment. This seems like an excellent book for me to start with. It is just I started taking my first algebraic topology course this semester, but I feel dull as every time I think about some possible theorem to prove, the stack-exchange community would just come up with bizarre (at least to me) counter-example in a short period that would take forever for me to verify.
$endgroup$
– Joe Martin
Apr 25 at 23:35
$begingroup$
Thank you very much for the comment. This seems like an excellent book for me to start with. It is just I started taking my first algebraic topology course this semester, but I feel dull as every time I think about some possible theorem to prove, the stack-exchange community would just come up with bizarre (at least to me) counter-example in a short period that would take forever for me to verify.
$endgroup$
– Joe Martin
Apr 25 at 23:35
$begingroup$
@Joe Martin, that's because they say it (or a similar counterexample) before. Don't judge your ability to conceive of brand new ideas against the experience of the entire MSE community.
$endgroup$
– Mark S.
Apr 26 at 11:20
$begingroup$
@Joe Martin, that's because they say it (or a similar counterexample) before. Don't judge your ability to conceive of brand new ideas against the experience of the entire MSE community.
$endgroup$
– Mark S.
Apr 26 at 11:20
add a comment |
$begingroup$
I think that it would indeed be odd for people to come up with exotic counterexamples to innocuous conjectures out of nowhere, as you say. Really, what is guiding those counterexamples is a lot of time and experience spent with problems and the material. When you are reading the statement of a theorem, try seeing what happens when you omit a hypothesis to see what may go wrong, and talk to people about it, either here online or in person to share your thoughts. The more you learn, the more connections you will make, and eventually you will begin to see more as you synthesize that knowledge.
answered Apr 25 at 22:27
Alex OrtizAlex Ortiz
11.8k21544
$endgroup$
add a comment |
$begingroup$
I think that it would indeed be odd for people to come up with exotic counterexamples to innocuous conjectures out of nowhere, as you say. Really, what is guiding those counterexamples is a lot of time and experience spent with problems and the material. When you are reading the statement of a theorem, try seeing what happens when you omit a hypothesis to see what may go wrong, and talk to people about it, either here online or in person to share your thoughts. The more you learn, the more connections you will make, and eventually you will begin to see more as you synthesize that knowledge.
answered Apr 25 at 22:27
Alex OrtizAlex Ortiz
11.8k21544
$endgroup$
add a comment |
$begingroup$
I think that it would indeed be odd for people to come up with exotic counterexamples to innocuous conjectures out of nowhere, as you say. Really, what is guiding those counterexamples is a lot of time and experience spent with problems and the material. When you are reading the statement of a theorem, try seeing what happens when you omit a hypothesis to see what may go wrong, and talk to people about it, either here online or in person to share your thoughts. The more you learn, the more connections you will make, and eventually you will begin to see more as you synthesize that knowledge.
answered Apr 25 at 22:27
Alex OrtizAlex Ortiz
11.8k21544
$endgroup$
I think that it would indeed be odd for people to come up with exotic counterexamples to innocuous conjectures out of nowhere, as you say. Really, what is guiding those counterexamples is a lot of time and experience spent with problems and the material. When you are reading the statement of a theorem, try seeing what happens when you omit a hypothesis to see what may go wrong, and talk to people about it, either here online or in person to share your thoughts. The more you learn, the more connections you will make, and eventually you will begin to see more as you synthesize that knowledge.
answered Apr 25 at 22:27
Alex OrtizAlex Ortiz
11.8k21544
answered Apr 25 at 22:27
Alex OrtizAlex Ortiz
11.8k21544
answered Apr 25 at 22:27
Alex OrtizAlex Ortiz
11.8k21544
answered Apr 25 at 22:27
Alex OrtizAlex Ortiz
11.8k21544
11.8k21544
add a comment |
add a comment |
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$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
Apr 25 at 22:43
$begingroup$
Another general question: How to attack “if true, prove it; if not true, give a counterexample” question?
$endgroup$
– YuiTo Cheng
Apr 26 at 3:58
$begingroup$
It can be difficult. When, in the latter 19th century, Weierstrass exhibited a continuous nowhere-differentiable $f:Bbb R to Bbb R$, many were surprised as many expected that to be impossible.
$endgroup$
– DanielWainfleet
Apr 26 at 11:24