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Alexandre Eremenko
Alexandre Eremenko
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A stronger result is true: there is no homeoorphismhomeomorphism of the plane which takes the spiral to the line. If the spiral is defined exactly as in your question (does not contain the origin) this is evident: your spiral in not a closed subset of the plane, while the line is.

But even if you add the origin to your definition of the spiral, the result still holds. Simply because the spiral with the origin added is not homeomorphic to the line.

A stronger result is true: there is no homeoorphism of the plane which takes the spiral to the line. If the spiral is defined exactly as in your question (does not contain the origin) this is evident: your spiral in not a closed subset of the plane, while the line is.

But even if you add the origin to your definition of the spiral, the result still holds. Simply because the spiral with the origin added is not homeomorphic to the line.

A stronger result is true: there is no homeomorphism of the plane which takes the spiral to the line. If the spiral is defined exactly as in your question (does not contain the origin) this is evident: your spiral in not a closed subset of the plane, while the line is.

But even if you add the origin to your definition of the spiral, the result still holds. Simply because the spiral with the origin added is not homeomorphic to the line.

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Alexandre Eremenko
Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429

A stronger result is true: there is no homeoorphism of the plane which takes the spiral to the line. If the spiral is defined exactly as in your question (does not contain the origin) this is evident: your spiral in not a closed subset of the plane, while the line is.

But even if you add the origin to your definition of the spiral, the result still holds. Simply because the spiral with the origin added is not homeomorphic to the line.