A plane curve which is nowhere an embedding

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Is there a continuous map $gamma colon [0,1] rightarrow mathbbR^2$ which satisfies the following?



  1. "Moves and never looks back": $gamma(0) = (0,0)$ and $gamma(t) neq (0,0)$ when $t neq 0$.


  2. "Is never simple": there is no choice of $0leq a < b leq 1$ making the restriction $gamma|_[a,b]$ injective.


I suspect the answer is "yes" just because otherwise would be too good to be true in the wild world of curves.






general-topology







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  • Can't you take $gamma(t) = (f[t]-f[0], 0)$ for $f$ a Weierstrass function?
    – user7530
    Sep 1 at 4:07










  • user7530 has reduced the problem to finding a function $gamma : [0,1] to mathbbR$ having the desired properties.
    – Paul Frost
    Sep 1 at 11:22










  • Another idea is to consider space filling curves. However, the usual examples do not satisy 1.
    – Paul Frost
    Sep 2 at 9:05














up vote
1
down vote

favorite












Is there a continuous map $gamma colon [0,1] rightarrow mathbbR^2$ which satisfies the following?



  1. "Moves and never looks back": $gamma(0) = (0,0)$ and $gamma(t) neq (0,0)$ when $t neq 0$.


  2. "Is never simple": there is no choice of $0leq a < b leq 1$ making the restriction $gamma|_[a,b]$ injective.


I suspect the answer is "yes" just because otherwise would be too good to be true in the wild world of curves.






general-topology







share|cite|improve this question





















  • Can't you take $gamma(t) = (f[t]-f[0], 0)$ for $f$ a Weierstrass function?
    – user7530
    Sep 1 at 4:07










  • user7530 has reduced the problem to finding a function $gamma : [0,1] to mathbbR$ having the desired properties.
    – Paul Frost
    Sep 1 at 11:22










  • Another idea is to consider space filling curves. However, the usual examples do not satisy 1.
    – Paul Frost
    Sep 2 at 9:05












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Is there a continuous map $gamma colon [0,1] rightarrow mathbbR^2$ which satisfies the following?



  1. "Moves and never looks back": $gamma(0) = (0,0)$ and $gamma(t) neq (0,0)$ when $t neq 0$.


  2. "Is never simple": there is no choice of $0leq a < b leq 1$ making the restriction $gamma|_[a,b]$ injective.


I suspect the answer is "yes" just because otherwise would be too good to be true in the wild world of curves.






general-topology







share|cite|improve this question













Is there a continuous map $gamma colon [0,1] rightarrow mathbbR^2$ which satisfies the following?



  1. "Moves and never looks back": $gamma(0) = (0,0)$ and $gamma(t) neq (0,0)$ when $t neq 0$.


  2. "Is never simple": there is no choice of $0leq a < b leq 1$ making the restriction $gamma|_[a,b]$ injective.


I suspect the answer is "yes" just because otherwise would be too good to be true in the wild world of curves.






general-topology




general-topology






share|cite|improve this question













share|cite|improve this question











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asked Sep 1 at 3:30









Cihan

1,640816




1,640816











  • Can't you take $gamma(t) = (f[t]-f[0], 0)$ for $f$ a Weierstrass function?
    – user7530
    Sep 1 at 4:07










  • user7530 has reduced the problem to finding a function $gamma : [0,1] to mathbbR$ having the desired properties.
    – Paul Frost
    Sep 1 at 11:22










  • Another idea is to consider space filling curves. However, the usual examples do not satisy 1.
    – Paul Frost
    Sep 2 at 9:05
















  • Can't you take $gamma(t) = (f[t]-f[0], 0)$ for $f$ a Weierstrass function?
    – user7530
    Sep 1 at 4:07










  • user7530 has reduced the problem to finding a function $gamma : [0,1] to mathbbR$ having the desired properties.
    – Paul Frost
    Sep 1 at 11:22










  • Another idea is to consider space filling curves. However, the usual examples do not satisy 1.
    – Paul Frost
    Sep 2 at 9:05















Can't you take $gamma(t) = (f[t]-f[0], 0)$ for $f$ a Weierstrass function?
– user7530
Sep 1 at 4:07




Can't you take $gamma(t) = (f[t]-f[0], 0)$ for $f$ a Weierstrass function?
– user7530
Sep 1 at 4:07












user7530 has reduced the problem to finding a function $gamma : [0,1] to mathbbR$ having the desired properties.
– Paul Frost
Sep 1 at 11:22




user7530 has reduced the problem to finding a function $gamma : [0,1] to mathbbR$ having the desired properties.
– Paul Frost
Sep 1 at 11:22












Another idea is to consider space filling curves. However, the usual examples do not satisy 1.
– Paul Frost
Sep 2 at 9:05




Another idea is to consider space filling curves. However, the usual examples do not satisy 1.
– Paul Frost
Sep 2 at 9:05















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