Pointed Gromov-Hausdorff Topological Stability for Non-compact Metric Spaces

Abstract

We combine the pointed Gromov-Hausdorff metric with the locally $C^0$-distance to obtain the pointed $C^0$-Gromov-Hausdorff distance between maps of possibly different non-compact pointed metric spaces. The latter is combined with Walters’s locally topological stability proposed by Lee–Nguyen–Yang, and GH-stability from Arbieto-Morales to obtain the notion of topologically GH-stable pointed homeomorphism. We give one example to show the difference between the distance when taking different base points in a pointed metric space.

Appendix A Appendix

Proposition A.1

(triangle inequality) Let $\left( X_{m}, x_{m}\right)$ be pointed metric spaces for $m=1,2,3$. If $d^p_{GH}\left( (X_{m}, x_{m}),(X_{n}, x_{n})\right) \le 1 / 2$ for $(m,n)=(1,2)$ and (2, 3), then

$$\begin{aligned} d^p_{GH}\left( (X_{1}, x_{1}),(X_{3}, x_{3})\right) \le 2\left[ d^p_{GH}\left( (X_{1}, x_{1}),(X_{2}, x_{2})\right) + d^p_{GH}\left( (X_{2}, x_{2}),(X_{3}, x_{3})\right) \right] \end{aligned}$$

Proof

Let $i_{mn}\in \textrm{App}_{\varepsilon _{mn}}\left( (X_{m}, x_{m}),(X_{n}, x_{n})\right)$, where (m, n) equal to (1, 2), or (2, 3). We use the notation $B^{m}$ for $B^{X_m}$, and $d^{m}$ for $d^{X_m}$, and take $\varepsilon _{mn}<\frac{1}{2}$. Then we have that $i_{mn}(x_m)=x_n$,

$$\begin{aligned} dis(i_{mn})|_{B^m(x_m,\varepsilon _{mn}^{-1})}<\varepsilon _{mn}, \quad B^n(x_n,\varepsilon _{mn}^{-1}-\varepsilon _{mn}) \subset N_{\varepsilon _{mn}}\left( i_{mn}(B^n(x_n, \varepsilon _{mn}^{-1}))\right) . \end{aligned}$$

We call $i_{13} = i_{23} \circ i_{12}$, and let $\varepsilon _{13}=2\left( \varepsilon _{12}+\varepsilon _{23}\right)$. We want to show that $i_{13}\in \textrm{App}_{\varepsilon _{13}}\left( (X_{1}, x_{1}),(X_{3}, x_{3})\right)$.

Observe that $\varepsilon _{mn}<2\varepsilon _{mn}< \varepsilon _{13}< 1$,

Let $p\in B^{1}\left( x_{1}, \varepsilon _{13}^{-1}\right) \subset B^{1}\left( x_{1}, \varepsilon _{12}^{-1}\right)$, and since $dis(i_{12})|_{B^1(x_1,\varepsilon _{12}^{-1})}<\varepsilon _{12}$ and $i_{12}(x_1)=x_2$ we have that

$$\begin{aligned} |d^1(p,x_1)-d^2(i_{12}(p),x_2)|\le \varepsilon _{12}, \end{aligned}$$

then

$$\begin{aligned} d^2(i_{12}(p),x_2)&\le d^1(p,x_1)+\varepsilon _{12}\\&\le \varepsilon _{13}^{-1}+\varepsilon _{12}\\&\le \frac{1+2\varepsilon _{12}(\varepsilon _{12} +\varepsilon _{23})}{2(\varepsilon _{12}+\varepsilon _{23})} \le \frac{1}{(\varepsilon _{12}+\varepsilon _{23})}\\&\le \frac{1}{\varepsilon _{23}}. \end{aligned}$$

Note that we use that $\varepsilon _{mn}<1/2$.

Now if $p_1,p_2\in B^{1}\left( x_{1}, \varepsilon _{13}^{-1}\right)$, $i_{12}(p_1),i_{12}(p_2)\in B^{2}\left( x_{2}, \varepsilon _{23}^{-1}\right)$, then

$$\begin{aligned}&|d^1(p_1,p_2)-d^3(i_{13}(p_1),i_{13}(p_2))|\\&\quad \le |d^1(p_1,p_2)-d^2(i_{12}(p_1),i_{12}(p_2))| +|d^2(i_{12}(p_1),i_{12}(p_2))-d^3(i_{13}(p_1),i_{13}(p_2))|\\&\quad \le \varepsilon _{12}+\varepsilon _{23}<\varepsilon _{13}. \end{aligned}$$

Then $dis(i_{13})|_{B^1(x_1,\varepsilon _{13}^{-1})} <\varepsilon _{13}$.

For the third part, let $p_3\in B^{3}\left( x_{3}, \varepsilon _{13}^{-1}-\varepsilon _{13}\right) \subset B^{3}\left( x_{3}, \varepsilon _{23}^{-1}\right) \subset N_{\varepsilon _{23}}\left( i_{23}B^2(x_2,\varepsilon _{23}^{-1})\right)$, then there exists $p_2\in B^2(x_2,\varepsilon _{23}^{-1})$, such that $d^3(i_{23}(p_2),p_3)<\varepsilon _{23}$, and using the distortion of $i_{23}$

$$\begin{aligned} d^{2}\left( p_{2}, x_{2}\right)&\le d^{3}(i_{23} (p_{2}), x_{3})+\varepsilon _{23} \\&\le d^{3}(i_{23}(p_{2}), p_{3}) +d^{3}(p_{3},x_3)+\varepsilon _{23}\\&\le \varepsilon _{13}^{-1}-\varepsilon _{13}+2\varepsilon _{23} =\varepsilon _{13}^{-1}-2\varepsilon _{12}\\&\le \varepsilon _{12}^{-1}-\varepsilon _{12}, \end{aligned}$$

that is, $p_{2} \in B^{2}\left( x_{2}, \varepsilon _{12}^{-1}-\varepsilon _{12}\right) \subset N_{\varepsilon _{12}}\left( i_{12}B^1(x_1,\varepsilon _{12}^{-1})\right)$. So there is $p_{1} \in B_{1}\left( x_{1}, \varepsilon _{12}^{-1}\right)$ such that $d^{2}\left( i_{12}\left( p_{1}\right) , p_{2}\right) <\varepsilon _{12},$ and again by the dilation of $i_{12}$

$$\begin{aligned} d^{1}\left( p_{1}, x_{1}\right)&\le d^{2}(i_{12}(p_{1}), p_{2})+d^{2}(p_{2}, x_{2})+\varepsilon _{12} \\&<\varepsilon _{12}+\varepsilon _{13}^{-1}-\varepsilon _{13} +2\varepsilon _{23}+\varepsilon _{12}\\&= \varepsilon _{13}^{-1}, \end{aligned}$$

that is $p_1\in B^1(x_1,\varepsilon _{13}^{-1})$. Now using the distortion of $i_{23}$ we have

$$\begin{aligned} d^{3}\left( i_{13}\left( p_{1}\right) , p_{3}\right)&\le d^{3}\left( i_{13}\left( p_{1}\right) , i_{23} \left( p_{2}\right) \right) +d^{3}\left( i_{23} \left( p_{2}\right) , p_{3}\right) \\&= d^{3}\left( i_{23}(i_{12}(p_{1})), i_{23}(p_{2})\right) +d^{3}\left( i_{23}(p_{2}), p_{3}\right) \\&\le d^{2}\left( i_{12}(p_{1}), p_{2}\right) +\varepsilon _{23}+d^{3}\left( i_{23}(p_{2}), p_{3}\right) \\&\le \varepsilon _{12}+2 \varepsilon _{23} \\&<\varepsilon _{13}. \end{aligned}$$

That is $B^3(x_3,\varepsilon _{13}^{-1}-\varepsilon _{13})\subset N_{\varepsilon _{13}}(i_{13}(B^1(x_1,\varepsilon _{13}^{-1})))$.

Making (m, n) equal to (2, 1), or (3, 2), and interchanging 1 by 3, we get that the same result for $i_{31}$, and the proof is complete. $\square$