Are spaces shaped like the digits 0, 8 and 9 homeomorphic topological spaces?












2















Consider the topological spaces shaped like the numerals "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
    2 hours ago






  • 1




    @GerryMyerson yeah! My mistake. Thank you.
    – Lucas Corrêa
    2 hours ago


















2















Consider the topological spaces shaped like the numerals "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
    2 hours ago






  • 1




    @GerryMyerson yeah! My mistake. Thank you.
    – Lucas Corrêa
    2 hours ago
















2












2








2








Consider the topological spaces shaped like the numerals "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 shaped like the numerals "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








edited 28 mins ago









Tanner Swett

3,9941638




3,9941638










asked 3 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
    2 hours ago






  • 1




    @GerryMyerson yeah! My mistake. Thank you.
    – Lucas Corrêa
    2 hours 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
    2 hours ago






  • 1




    @GerryMyerson yeah! My mistake. Thank you.
    – Lucas Corrêa
    2 hours 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
2 hours 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
2 hours ago




1




1




@GerryMyerson yeah! My mistake. Thank you.
– Lucas Corrêa
2 hours ago






@GerryMyerson yeah! My mistake. Thank you.
– Lucas Corrêa
2 hours 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 $f:Xto Y$ 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
    2 hours ago










  • I have made an edit to your post. Feel free to revert to its original version if you disagree with my edit.
    – user 170039
    29 mins ago











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1 Answer
1






active

oldest

votes








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 $f:Xto Y$ 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
    2 hours ago










  • I have made an edit to your post. Feel free to revert to its original version if you disagree with my edit.
    – user 170039
    29 mins 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 $f:Xto Y$ 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
    2 hours ago










  • I have made an edit to your post. Feel free to revert to its original version if you disagree with my edit.
    – user 170039
    29 mins 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 $f:Xto Y$ 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 $f:Xto Y$ 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








edited 29 mins ago









user 170039

10.4k42465




10.4k42465










answered 2 hours ago









William Elliot

7,2382519




7,2382519












  • Nice! Thanks for the hint!
    – Lucas Corrêa
    2 hours ago










  • I have made an edit to your post. Feel free to revert to its original version if you disagree with my edit.
    – user 170039
    29 mins ago


















  • Nice! Thanks for the hint!
    – Lucas Corrêa
    2 hours ago










  • I have made an edit to your post. Feel free to revert to its original version if you disagree with my edit.
    – user 170039
    29 mins ago
















Nice! Thanks for the hint!
– Lucas Corrêa
2 hours ago




Nice! Thanks for the hint!
– Lucas Corrêa
2 hours ago












I have made an edit to your post. Feel free to revert to its original version if you disagree with my edit.
– user 170039
29 mins ago




I have made an edit to your post. Feel free to revert to its original version if you disagree with my edit.
– user 170039
29 mins ago


















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