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Let $f:E\to F$ be a function between sets in Euclidean space.

(1) If $f$ is Lipschitz on $E$, then dim $f(E)$ $\leq$ dim $E$.

(2) If $f$ is a bi-Lipschitz bijection on $E$, then dim $E$=dim $f(E)$.

My question is the follwoing:

Are there some examples such that dim $E$$\not=$dim $f(E)$ ($E\subset \mathbb{R}$) if $f:\mathbb{R}\to \mathbb{R}$ is a homeomorphism?

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Many Cantor sets are homeomorphic (Homeomorphism between two Cantor sets) and they will not have the same Hausdorff dimension (Hausdorff dimension of Cantor set).

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  • $\begingroup$ Thank you so much. Moreover, are there some examples such that dim $E$ $\not=$dim $f(E)$ ($E\subset\mathbb{R}$) if $f:\mathbb{R}\to\mathbb{R}$ is a homeomorphism? $\endgroup$ – Adele Nov 30 '19 at 14:35
  • $\begingroup$ Well, take the examples in my answer, the homeomorphism extends to the whole real line (we can pick a homeomorphism mapping the excluded intervals of one Cantor set to the excluded ones of the other Cantor set). By the way if you change the question, you should rather add what you is the additional information you'd like or open a new question. In this case it is not so bad, but another time the answer could be completely off and this would confuse people reading this. $\endgroup$ – Severin Schraven Nov 30 '19 at 14:58

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