Why do we study Cantor Set?

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For finding counter examples. That does not sound convincing enough, at least not always. Why as a object in its own right the study of Cantor Set has merit ?






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    For finding counter examples. That does not sound convincing enough, at least not always. Why as a object in its own right the study of Cantor Set has merit ?






    mathematical-analysis







    share|improve this question























      up vote
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      up vote
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      down vote

      favorite











      For finding counter examples. That does not sound convincing enough, at least not always. Why as a object in its own right the study of Cantor Set has merit ?






      mathematical-analysis







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      For finding counter examples. That does not sound convincing enough, at least not always. Why as a object in its own right the study of Cantor Set has merit ?






      mathematical-analysis




      mathematical-analysis






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      asked 7 hours ago









      Vagabond

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          1. in beginning real analysis: to counter the naive notion that a "closed set" is a union of closed intervals, plus a few single points.


          2. In beginning Lebesgue measure: the easiest example of an uncountable set of measure zero


          3. In general topology: sets homeomorphic to the Cantor set are useful in proofs


          4. Fractal geometry: many fractals are homeomorphic to the Cantor set. Mandelbrot calls such a thing a Cantor dust to suggest its appearance.






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          • Can one claim and substantiate that the study of cantor set is something fundamental and not just a source of ready counterexample ?
            – Vagabond
            1 hour ago

















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          I think that you may be selling short the value of a counterexample! They are quite useful for making sure that you have not proven too much. i.e. When you have a plausible but only semi-formal argument, how do you tell if it is worth the effort in making it rigorous? Checking against counterexamples is often a useful step.



          Still, there are relations to unbounded paths in an infinite binary tree without leaves. That is, suppose that you start at the root and write $0.$, thought of as a ternary number. As we travel to the left or right, write a $0$ or $2$, respectively, for the following digit. Continuing on we get a ternary expansion defining a real number in the Cantor set. Correspondingly, the ternary expansion of an element of the Cantor set gives rise to a path from the root in the tree.





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            1. in beginning real analysis: to counter the naive notion that a "closed set" is a union of closed intervals, plus a few single points.


            2. In beginning Lebesgue measure: the easiest example of an uncountable set of measure zero


            3. In general topology: sets homeomorphic to the Cantor set are useful in proofs


            4. Fractal geometry: many fractals are homeomorphic to the Cantor set. Mandelbrot calls such a thing a Cantor dust to suggest its appearance.






            share|improve this answer




















            • Can one claim and substantiate that the study of cantor set is something fundamental and not just a source of ready counterexample ?
              – Vagabond
              1 hour ago














            up vote
            3
            down vote













            1. in beginning real analysis: to counter the naive notion that a "closed set" is a union of closed intervals, plus a few single points.


            2. In beginning Lebesgue measure: the easiest example of an uncountable set of measure zero


            3. In general topology: sets homeomorphic to the Cantor set are useful in proofs


            4. Fractal geometry: many fractals are homeomorphic to the Cantor set. Mandelbrot calls such a thing a Cantor dust to suggest its appearance.






            share|improve this answer




















            • Can one claim and substantiate that the study of cantor set is something fundamental and not just a source of ready counterexample ?
              – Vagabond
              1 hour ago












            up vote
            3
            down vote










            up vote
            3
            down vote









            1. in beginning real analysis: to counter the naive notion that a "closed set" is a union of closed intervals, plus a few single points.


            2. In beginning Lebesgue measure: the easiest example of an uncountable set of measure zero


            3. In general topology: sets homeomorphic to the Cantor set are useful in proofs


            4. Fractal geometry: many fractals are homeomorphic to the Cantor set. Mandelbrot calls such a thing a Cantor dust to suggest its appearance.






            share|improve this answer












            1. in beginning real analysis: to counter the naive notion that a "closed set" is a union of closed intervals, plus a few single points.


            2. In beginning Lebesgue measure: the easiest example of an uncountable set of measure zero


            3. In general topology: sets homeomorphic to the Cantor set are useful in proofs


            4. Fractal geometry: many fractals are homeomorphic to the Cantor set. Mandelbrot calls such a thing a Cantor dust to suggest its appearance.







            share|improve this answer












            share|improve this answer



            share|improve this answer










            answered 1 hour ago









            Gerald Edgar

            3,04411013




            3,04411013











            • Can one claim and substantiate that the study of cantor set is something fundamental and not just a source of ready counterexample ?
              – Vagabond
              1 hour ago
















            • Can one claim and substantiate that the study of cantor set is something fundamental and not just a source of ready counterexample ?
              – Vagabond
              1 hour ago















            Can one claim and substantiate that the study of cantor set is something fundamental and not just a source of ready counterexample ?
            – Vagabond
            1 hour ago




            Can one claim and substantiate that the study of cantor set is something fundamental and not just a source of ready counterexample ?
            – Vagabond
            1 hour ago










            up vote
            0
            down vote













            I think that you may be selling short the value of a counterexample! They are quite useful for making sure that you have not proven too much. i.e. When you have a plausible but only semi-formal argument, how do you tell if it is worth the effort in making it rigorous? Checking against counterexamples is often a useful step.



            Still, there are relations to unbounded paths in an infinite binary tree without leaves. That is, suppose that you start at the root and write $0.$, thought of as a ternary number. As we travel to the left or right, write a $0$ or $2$, respectively, for the following digit. Continuing on we get a ternary expansion defining a real number in the Cantor set. Correspondingly, the ternary expansion of an element of the Cantor set gives rise to a path from the root in the tree.





            share
























              up vote
              0
              down vote













              I think that you may be selling short the value of a counterexample! They are quite useful for making sure that you have not proven too much. i.e. When you have a plausible but only semi-formal argument, how do you tell if it is worth the effort in making it rigorous? Checking against counterexamples is often a useful step.



              Still, there are relations to unbounded paths in an infinite binary tree without leaves. That is, suppose that you start at the root and write $0.$, thought of as a ternary number. As we travel to the left or right, write a $0$ or $2$, respectively, for the following digit. Continuing on we get a ternary expansion defining a real number in the Cantor set. Correspondingly, the ternary expansion of an element of the Cantor set gives rise to a path from the root in the tree.





              share






















                up vote
                0
                down vote










                up vote
                0
                down vote









                I think that you may be selling short the value of a counterexample! They are quite useful for making sure that you have not proven too much. i.e. When you have a plausible but only semi-formal argument, how do you tell if it is worth the effort in making it rigorous? Checking against counterexamples is often a useful step.



                Still, there are relations to unbounded paths in an infinite binary tree without leaves. That is, suppose that you start at the root and write $0.$, thought of as a ternary number. As we travel to the left or right, write a $0$ or $2$, respectively, for the following digit. Continuing on we get a ternary expansion defining a real number in the Cantor set. Correspondingly, the ternary expansion of an element of the Cantor set gives rise to a path from the root in the tree.





                share












                I think that you may be selling short the value of a counterexample! They are quite useful for making sure that you have not proven too much. i.e. When you have a plausible but only semi-formal argument, how do you tell if it is worth the effort in making it rigorous? Checking against counterexamples is often a useful step.



                Still, there are relations to unbounded paths in an infinite binary tree without leaves. That is, suppose that you start at the root and write $0.$, thought of as a ternary number. As we travel to the left or right, write a $0$ or $2$, respectively, for the following digit. Continuing on we get a ternary expansion defining a real number in the Cantor set. Correspondingly, the ternary expansion of an element of the Cantor set gives rise to a path from the root in the tree.






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                answered 6 mins ago









                Adam

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