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Topology

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Metric Space Of Cardinality At Least 4

TOPO-4TW1N9

Suppose that $X$ is a set containing at least four different points, and $d: X \times X\to \mathbb{R}$ is a real-valued function. You are told that $d$ is a metric on $X$.

Choose ALL the correct statements from the list.

A

the null-set $\{(x,y)\in X\times X: d(x,y) = 0\}$ of $d$ contains a single element.

B

$d$ is a symmetric function.

C

the null-set $\{(x,y)\in X\times X: d(x,y) = 0\}$ of $d$ is empty.

D

$d$ is not onto.

E

if $x,y,z,v$ are four different points in $X$ such that $d(x,y)\leq d(y,z)\leq d(z,v)$ then $d(x,y) \leq d(x,v)$.

F

if $x,y,z$ are three different points in $X$ such that $d(x,y)= d(y,z)$, then $d(x,z)\leq \frac{3}{2} d(z,y)$.