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










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  • $begingroup$
    This article addresses the general questions that surround yours: On teaching mathematics
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    – avs
    4 hours ago
















1












$begingroup$


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










share|cite|improve this question







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.







$endgroup$












  • $begingroup$
    This article addresses the general questions that surround yours: On teaching mathematics
    $endgroup$
    – avs
    4 hours ago














1












1








1





$begingroup$


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










share|cite|improve this question







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






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Joe Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question







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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Joe Martin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 4 hours ago









Joe MartinJoe Martin

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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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Check out our Code of Conduct.












  • $begingroup$
    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




$begingroup$
This article addresses the general questions that surround yours: On teaching mathematics
$endgroup$
– avs
4 hours ago










2 Answers
2






active

oldest

votes


















2












$begingroup$

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.






share|cite|improve this answer









$endgroup$









  • 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



















3












$begingroup$

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.






share|cite|improve this answer









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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$









    • 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
















    2












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$









    • 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














    2












    2








    2





    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$



    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.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    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














    • 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











    3












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$


















      3












      $begingroup$

      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.






      share|cite|improve this answer









      $endgroup$
















        3












        3








        3





        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$



        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 4 hours ago









        Alex OrtizAlex Ortiz

        11.6k21442




        11.6k21442






















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