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Topology

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Examples of Proper Maps

TOPO-7VTVMD

Which of the following map $f$ from $X$ to $Y$ is (are) proper?

A

$X$ is a metric space, $Y=\mathbb{R}$, $a\in X$, $f(x)=d(x,a)$.

B

$X=Y=\mathbb{R}$, $f(x)=x^5+2x^4-x^3+1$.

C

$X=\mathbb{Z}$ with cofinite topology, $Y=\mathbb{Z}$ with discrete topology, and $f(n)=$ largest prime factor of $n$.

D

$X=\mathbb{R}^3$, $Y=\mathbb{R}$, $f((a,b,c))=a^2+b^2+c^2$.