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Corrected bad notation of objects
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Evgeny
Evgeny
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Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 + y^2 \leqslant C \cdot z^2 \rbrace $. Also we have a homeomorphism $h$ that maps $B$ on itself ($h(B) = B$) and $h(0) = 0$ (if it's crucial, $h(B \bigcap Oz) = B \bigcap Oz$$h(B \cap Oz) = B \cap Oz$ and $h(0) = 0$). So, the question is: if we take a cone, corresponding to some value $C^*$$C_1$, does exist some $C'$$C_2$ that $f(X_{C^*}) \subseteq X_{C'}$$f(X_{C_1}) \subseteq X_{C_2}$?

Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 + y^2 \leqslant C \cdot z^2 \rbrace $. Also we have a homeomorphism $h$ that maps $B$ on itself ($h(B) = B$) and $h(0) = 0$ (if it's crucial, $h(B \bigcap Oz) = B \bigcap Oz$ and $h(0) = 0$). So, the question is: if we take a cone, corresponding to some value $C^*$, does exist some $C'$ that $f(X_{C^*}) \subseteq X_{C'}$?

Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 + y^2 \leqslant C \cdot z^2 \rbrace $. Also we have a homeomorphism $h$ that maps $B$ on itself ($h(B) = B$) and $h(0) = 0$ (if it's crucial, $h(B \cap Oz) = B \cap Oz$ and $h(0) = 0$). So, the question is: if we take a cone, corresponding to some value $C_1$, does exist some $C_2$ that $f(X_{C_1}) \subseteq X_{C_2}$?

added 20 characters in body
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Evgeny
Evgeny
  • 165
  • 6

Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 + y^2 \leqslant C \cdot z^2 \rbrace $. Also we have a homeomorphism $h$ that maps $B$ on itself ($h(B) = B$) and $h(0) = 0$ (if it's crucial, $h(Oz) = Oz$$h(B \bigcap Oz) = B \bigcap Oz$ and $h(0) = 0$). So, the question is: if we take a cone, corresponding to some value $C^*$, does exist some $C'$ that $f(X_{C^*}) \subseteq X_{C'}$?

Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 + y^2 \leqslant C \cdot z^2 \rbrace $. Also we have a homeomorphism $h$ that maps $B$ on itself ($h(B) = B$) and $h(0) = 0$ (if it's crucial, $h(Oz) = Oz$ and $h(0) = 0$). So, the question is: if we take a cone, corresponding to some value $C^*$, does exist some $C'$ that $f(X_{C^*}) \subseteq X_{C'}$?

Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 + y^2 \leqslant C \cdot z^2 \rbrace $. Also we have a homeomorphism $h$ that maps $B$ on itself ($h(B) = B$) and $h(0) = 0$ (if it's crucial, $h(B \bigcap Oz) = B \bigcap Oz$ and $h(0) = 0$). So, the question is: if we take a cone, corresponding to some value $C^*$, does exist some $C'$ that $f(X_{C^*}) \subseteq X_{C'}$?

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Evgeny
Evgeny
  • 165
  • 6

Does homeomorphism preserves the family of cones?

Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 + y^2 \leqslant C \cdot z^2 \rbrace $. Also we have a homeomorphism $h$ that maps $B$ on itself ($h(B) = B$) and $h(0) = 0$ (if it's crucial, $h(Oz) = Oz$ and $h(0) = 0$). So, the question is: if we take a cone, corresponding to some value $C^*$, does exist some $C'$ that $f(X_{C^*}) \subseteq X_{C'}$?