All quasi-isometry is coarsely surjective

eraldcoil · **Posted:** Thu, 8 Nov 2018 07:08:33 UTC

Definition. (X,d_X),(Y,d_Y)

metric spaces.
$f:X\to Y$ is quasi-isometry if

$d_Y(f(x),f(x'))\leq Ld_X(x,x')+C$

exists coarse inverse, i.e. $\overline{f}:Y\to X$ such that

$d_X(\overline{f}f(x),x)\leq C$

$d_Y(f\overline{f}(y),y)\leq C$

and $d_X(\overline{f}(y),\overline{f}(y'))\leq Ld_Y(y,y')+C$

for all $x,x'\in X$ , for all $y,y'\in Y$

How prove that $f:X\to Y$ ( L-C

) quasi-isometry then

is coarsely surjective? i.e. $\forall y\in Y$ , $\exists f(x)\in f(X)$ such that $d_Y(y,f(x))\leq C$

S.O.S. Mathematics CyberBoard

All quasi-isometry is coarsely surjective

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