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 $f(X\_{C\_1}) \subseteq X\_{C\_2}$? 



Does homeomorphism preserves the family of cones?
Here is the counterexample, re-expressed. Let $S\_r$ be the sphere of radius $r \epsilon$. 
Construct a homeomorphism $h : B \to B$ with the following properties: 


* $h(S\_r) = S\_r$ for each $r \in [0,1]$
* $h(S\_{1/3} \cap X\_{C^\*}) = S\_{1/3} \cap X\_{C^\*}$
* $h(S\_{2/3} \cap X\_{C^\*}) \supset S\_{2/3} \cap (\mathbb{R}^3 - X\_{C^\*})$
* And, if you like, $h$ preserves the $z$-axis, which prevents the last $\supset$ from being $=$.


