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# Proper Maps and Metric Spaces

TOPO-WXCYIM

Which of the following statements is(are) true?

A

If $X$ is a locally compact metric space with metric $d$, $a\in\mathbb{X}$, then $x\mapsto d(a,x)$ is proper.

B

$X$ is a metric space with metric $d$, if $x\mapsto d(a,x)$ is proper for some $a\in\mathbb{X}$, then $X$ is locally compact.

C

Let $X$ and $Y$ be two locally compact metric spaces, with metric $d_X$ and $d_Y$. If continuous map $f: X\rightarrow Y$ is proper, then $f$ is a quasi-isometry ($\exists C, b$ such that $\forall x,y\in\mathbb{X}$, ${1\over C}d_X(x,y)-b\leq d_Y(f(x),f(y))\leq Cd_X(x,y)+b$.

D

Let $X$ and $Y$ be two locally compact metric spaces, with metric $d_X$ and $d_Y$. If continuous map $f: X\rightarrow Y$ is a quasiisometry then it is proper.