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I am reading thorugh some topological definitions, in my book it is stated that $id_M:(M,\tau_d)\rightarrow(M,\tau_h),x\rightarrow x$ is a Homeomorphism where

$(M,d)$ is a metric space, $h(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric $h$ on $M$.

A Homeomorphism between topological spaces is a map which is bijective, continuous and the inverse map is also continuous.

I would really appreciate it if could help me showing this properties.

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  • $\begingroup$ note that $h\leq d$ $\endgroup$ – Federica Maggioni May 6 '13 at 18:34
  • $\begingroup$ Bijectiveness is clear. Continuities can be proven by showing that $\tau_d=\tau_h$. $\endgroup$ – Damian Sobota May 6 '13 at 18:35
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Suppose $x_n \to x$ under $d$, i.e. $d(x_n,x) \to 0$. Then $h(x_n,x) = \frac{d(x_n,x)}{1+d(x_n,x)} \to 0$ (that's pretty easy to show). Therefore, $id : (M,d) \to (M,h)$ is continuous. It is clearly a bijection, and one can easily show that $h(x_n,x) \to 0 \Rightarrow d(x_n,x) \to 0$. Hence, the inverse is continuous, and we have thus shown that $id$ is a homeomorphism.

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HINT: $B_d(x,\epsilon)=B_h\left(x,\frac{\epsilon}{1+\epsilon}\right)$. You may find it useful to observe that the function $f(x)=\frac{x}{1+x}$ is monotone increasing on $[0,\to)$.

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