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Francesco Polizzi
Francesco Polizzi
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Given a homeomorphism on R^3$\mathbb{R}^3$, can its effects on a compact subset always be realized withby a homeomorphism that's non-identity only on a compact set?

Let $f_1 : \mathbb{R}^3 \to \mathbb{R}^3$$f_1 \colon \mathbb{R}^3 \to \mathbb{R}^3$ be a homeomorphism, and let $K_1 \subseteq \mathbb{R}^3$ be compact. Does there always exist a homeomorphism $f_2 : \mathbb{R}^3 \to \mathbb{R}^3$$f_2 \colon \mathbb{R}^3 \to \mathbb{R}^3$ and a compact $K_2 \subseteq \mathbb{R}^3$ such that

  1. $K_1 \subseteq K_2$
  2. $f_1(K_1) = f_2(K_1)$
  3. $f_2$ is identity on $\mathbb{R}^3 - K_2^\circ$?

(Hypothesis 1 is probably unnecessary in light of (2) and (3), I just thought it might be clearer to include)

Given a homeomorphism on R^3, can its effects on a compact subset always be realized with a homeomorphism that's non-identity only on a compact set?

Let $f_1 : \mathbb{R}^3 \to \mathbb{R}^3$ be a homeomorphism, and let $K_1 \subseteq \mathbb{R}^3$ be compact. Does there always exist a homeomorphism $f_2 : \mathbb{R}^3 \to \mathbb{R}^3$ and a compact $K_2 \subseteq \mathbb{R}^3$ such that

  1. $K_1 \subseteq K_2$
  2. $f_1(K_1) = f_2(K_1)$
  3. $f_2$ is identity on $\mathbb{R}^3 - K_2^\circ$?

(Hypothesis 1 is probably unnecessary in light of (2) and (3), I just thought it might be clearer to include)

Given a homeomorphism on $\mathbb{R}^3$, can its effects on a compact subset be realized by a homeomorphism that's non-identity only on a compact set?

Let $f_1 \colon \mathbb{R}^3 \to \mathbb{R}^3$ be a homeomorphism, and let $K_1 \subseteq \mathbb{R}^3$ be compact. Does there always exist a homeomorphism $f_2 \colon \mathbb{R}^3 \to \mathbb{R}^3$ and a compact $K_2 \subseteq \mathbb{R}^3$ such that

  1. $K_1 \subseteq K_2$
  2. $f_1(K_1) = f_2(K_1)$
  3. $f_2$ is identity on $\mathbb{R}^3 - K_2^\circ$?

(Hypothesis 1 is probably unnecessary in light of (2) and (3), I just thought it might be clearer to include)

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cloudman123
cloudman123
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Given a homeomorphism on R^3, can its effects on a compact subset always be realized with a homeomorphism that's non-identity only on a compact set?

Let $f_1 : \mathbb{R}^3 \to \mathbb{R}^3$ be a homeomorphism, and let $K_1 \subseteq \mathbb{R}^3$ be compact. Does there always exist a homeomorphism $f_2 : \mathbb{R}^3 \to \mathbb{R}^3$ and a compact $K_2 \subseteq \mathbb{R}^3$ such that

  1. $K_1 \subseteq K_2$
  2. $f_1(K_1) = f_2(K_1)$
  3. $f_2$ is identity on $\mathbb{R}^3 - K_2^\circ$?

(Hypothesis 1 is probably unnecessary in light of (2) and (3), I just thought it might be clearer to include)