How to be good at coming up with counter example in Topology
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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 4 hours ago
Joe MartinJoe Martin
185
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Joe Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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 4 hours ago
Joe MartinJoe Martin
185
New contributor
Joe Martin 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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This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
4 hours ago
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 4 hours ago
Joe MartinJoe Martin
185
New contributor
Joe Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
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 4 hours ago
Joe MartinJoe Martin
185
New contributor
Joe Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Joe MartinJoe Martin
185
New contributor
Joe Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Joe MartinJoe Martin
185
New contributor
Joe Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Joe MartinJoe Martin
185
asked 4 hours ago
Joe MartinJoe Martin
185
185
New contributor
Joe Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Joe Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Joe Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
4 hours ago
add a comment |
$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
4 hours ago
$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
4 hours ago
$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
4 hours ago
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 4 hours ago
avsavs
4,4151515
$endgroup$
1
$begingroup$
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
4 hours ago
$begingroup$
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
3 hours ago
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 4 hours ago
Alex OrtizAlex Ortiz
11.6k21442
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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 4 hours ago
avsavs
4,4151515
$endgroup$
1
$begingroup$
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
4 hours ago
$begingroup$
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
3 hours ago
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 4 hours ago
avsavs
4,4151515
$endgroup$
1
$begingroup$
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
4 hours ago
$begingroup$
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
3 hours ago
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 4 hours ago
avsavs
4,4151515
$endgroup$
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 4 hours ago
avsavs
4,4151515
answered 4 hours ago
avsavs
4,4151515
answered 4 hours ago
avsavs
4,4151515
answered 4 hours ago
avsavs
4,4151515
4,4151515
1
$begingroup$
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
4 hours ago
$begingroup$
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
3 hours ago
add a comment |
1
$begingroup$
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
4 hours ago
$begingroup$
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
3 hours ago
1
1
$begingroup$
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
4 hours ago
$begingroup$
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
4 hours ago
$begingroup$
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
3 hours ago
$begingroup$
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
3 hours ago
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 4 hours ago
Alex OrtizAlex Ortiz
11.6k21442
$endgroup$
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 4 hours ago
Alex OrtizAlex Ortiz
11.6k21442
$endgroup$
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 4 hours ago
Alex OrtizAlex Ortiz
11.6k21442
$endgroup$
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 4 hours ago
Alex OrtizAlex Ortiz
11.6k21442
answered 4 hours ago
Alex OrtizAlex Ortiz
11.6k21442
answered 4 hours ago
Alex OrtizAlex Ortiz
11.6k21442
answered 4 hours ago
Alex OrtizAlex Ortiz
11.6k21442
11.6k21442
add a comment |
add a comment |
Joe Martin is a new contributor. Be nice, and check out our Code of Conduct.
Joe Martin is a new contributor. Be nice, and check out our Code of Conduct.
Joe Martin is a new contributor. Be nice, and check out our Code of Conduct.
Joe Martin is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
4 hours ago