# Discrete and metric topologies equivalence

Given a set $X$, define a function $d:X\times X\rightarrow \mathbb{R}$ by $d(x,y) = 1$ if $x\neq y$ and $d(x,y)=0$ if $x=y$. Show that the metric topology on $X$ is equal to the discrete topology.

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Can you add the (homework) tag please, and maybe say something about what you've tried in solving this? – J. M. Apr 30 '11 at 1:13
What a discrete hint! – ncmathsadist Apr 30 '11 at 2:51
The title does not reflect the question. – lhf Apr 30 '11 at 3:00
$d$ should take values in $\mathbb R$, not $\mathbb R^2$. – lhf Apr 30 '11 at 3:01

Hint: What does the ball of radius $1/2$ around $x$ look like?