Are 0,8 and 9 homeomorphic topological space?












2















Consider the topological spaces "0","8" and "9" in $mathbb{R}^{2}$. Are they homeomorphic?




I have an approach that doesnt look very rigorous to me. I wanted to know how to formalize this if its correct.




  • 0 and 8 are not homeomorphic since excluding one point of 0 the space is still connected, but excluding the "tangent point" of 8, we have a disconnected space.


  • Same idea for 8 and 9.


  • The space 9 is union of one circle and one arc. The arc is homeomorphic to the circle, so we can view 9 as a union of two circles, then 8 and 9 are homeomorphic



PS: the topology of the spaces is induced by topology of $mathbb{R}^{2}$.






general-topology metric-spaces







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  • 2




    "The arc is homeomorphic to the circle." No, it isn't. An arc is disconnected by removing a point; a circle isn't.
    – Gerry Myerson
    1 hour ago






  • 1




    @GerryMyerson yeah! My mistake. Thank you.
    – Lucas Corrêa
    1 hour ago


















2















Consider the topological spaces "0","8" and "9" in $mathbb{R}^{2}$. Are they homeomorphic?




I have an approach that doesnt look very rigorous to me. I wanted to know how to formalize this if its correct.




  • 0 and 8 are not homeomorphic since excluding one point of 0 the space is still connected, but excluding the "tangent point" of 8, we have a disconnected space.


  • Same idea for 8 and 9.


  • The space 9 is union of one circle and one arc. The arc is homeomorphic to the circle, so we can view 9 as a union of two circles, then 8 and 9 are homeomorphic



PS: the topology of the spaces is induced by topology of $mathbb{R}^{2}$.






general-topology metric-spaces







share|cite|improve this question


















  • 2




    "The arc is homeomorphic to the circle." No, it isn't. An arc is disconnected by removing a point; a circle isn't.
    – Gerry Myerson
    1 hour ago






  • 1




    @GerryMyerson yeah! My mistake. Thank you.
    – Lucas Corrêa
    1 hour ago
















2












2








2








Consider the topological spaces "0","8" and "9" in $mathbb{R}^{2}$. Are they homeomorphic?




I have an approach that doesnt look very rigorous to me. I wanted to know how to formalize this if its correct.




  • 0 and 8 are not homeomorphic since excluding one point of 0 the space is still connected, but excluding the "tangent point" of 8, we have a disconnected space.


  • Same idea for 8 and 9.


  • The space 9 is union of one circle and one arc. The arc is homeomorphic to the circle, so we can view 9 as a union of two circles, then 8 and 9 are homeomorphic



PS: the topology of the spaces is induced by topology of $mathbb{R}^{2}$.






general-topology metric-spaces







share|cite|improve this question














Consider the topological spaces "0","8" and "9" in $mathbb{R}^{2}$. Are they homeomorphic?




I have an approach that doesnt look very rigorous to me. I wanted to know how to formalize this if its correct.




  • 0 and 8 are not homeomorphic since excluding one point of 0 the space is still connected, but excluding the "tangent point" of 8, we have a disconnected space.


  • Same idea for 8 and 9.


  • The space 9 is union of one circle and one arc. The arc is homeomorphic to the circle, so we can view 9 as a union of two circles, then 8 and 9 are homeomorphic



PS: the topology of the spaces is induced by topology of $mathbb{R}^{2}$.






general-topology metric-spaces




general-topology metric-spaces






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 hours ago









Lucas Corrêa

1,4661321




1,4661321








  • 2




    "The arc is homeomorphic to the circle." No, it isn't. An arc is disconnected by removing a point; a circle isn't.
    – Gerry Myerson
    1 hour ago






  • 1




    @GerryMyerson yeah! My mistake. Thank you.
    – Lucas Corrêa
    1 hour ago
















  • 2




    "The arc is homeomorphic to the circle." No, it isn't. An arc is disconnected by removing a point; a circle isn't.
    – Gerry Myerson
    1 hour ago






  • 1




    @GerryMyerson yeah! My mistake. Thank you.
    – Lucas Corrêa
    1 hour ago










2




2




"The arc is homeomorphic to the circle." No, it isn't. An arc is disconnected by removing a point; a circle isn't.
– Gerry Myerson
1 hour ago




"The arc is homeomorphic to the circle." No, it isn't. An arc is disconnected by removing a point; a circle isn't.
– Gerry Myerson
1 hour ago




1




1




@GerryMyerson yeah! My mistake. Thank you.
– Lucas Corrêa
1 hour ago






@GerryMyerson yeah! My mistake. Thank you.
– Lucas Corrêa
1 hour ago












1 Answer
1






active

oldest

votes


















8














0 has no cut points.

8 has exactly one cut point.

9 has infinitely many cutpoints.

To show there are no homeomorphisms

among 0,8,9 use the exercise.



Exercise. Prove if f:X -> Y is homeomorphism and

p cutpoint of X, then f(p) is cutpoint of Y.



Also show an arc is not homeomorphic to a circle.






share|cite|improve this answer





















  • Nice! Thanks for the hint!
    – Lucas Corrêa
    1 hour ago











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1 Answer
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active

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









8














0 has no cut points.

8 has exactly one cut point.

9 has infinitely many cutpoints.

To show there are no homeomorphisms

among 0,8,9 use the exercise.



Exercise. Prove if f:X -> Y is homeomorphism and

p cutpoint of X, then f(p) is cutpoint of Y.



Also show an arc is not homeomorphic to a circle.






share|cite|improve this answer





















  • Nice! Thanks for the hint!
    – Lucas Corrêa
    1 hour ago
















8














0 has no cut points.

8 has exactly one cut point.

9 has infinitely many cutpoints.

To show there are no homeomorphisms

among 0,8,9 use the exercise.



Exercise. Prove if f:X -> Y is homeomorphism and

p cutpoint of X, then f(p) is cutpoint of Y.



Also show an arc is not homeomorphic to a circle.






share|cite|improve this answer





















  • Nice! Thanks for the hint!
    – Lucas Corrêa
    1 hour ago














8












8








8






0 has no cut points.

8 has exactly one cut point.

9 has infinitely many cutpoints.

To show there are no homeomorphisms

among 0,8,9 use the exercise.



Exercise. Prove if f:X -> Y is homeomorphism and

p cutpoint of X, then f(p) is cutpoint of Y.



Also show an arc is not homeomorphic to a circle.






share|cite|improve this answer












0 has no cut points.

8 has exactly one cut point.

9 has infinitely many cutpoints.

To show there are no homeomorphisms

among 0,8,9 use the exercise.



Exercise. Prove if f:X -> Y is homeomorphism and

p cutpoint of X, then f(p) is cutpoint of Y.



Also show an arc is not homeomorphic to a circle.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 1 hour ago









William Elliot

7,2382519




7,2382519












  • Nice! Thanks for the hint!
    – Lucas Corrêa
    1 hour ago


















  • Nice! Thanks for the hint!
    – Lucas Corrêa
    1 hour ago
















Nice! Thanks for the hint!
– Lucas Corrêa
1 hour ago




Nice! Thanks for the hint!
– Lucas Corrêa
1 hour ago


















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