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.
Discrete and metric topologies equivalence
1
$\begingroup$
general-topology
-
0The title does not reflect the question. – 2011-04-30
1 Answers
5
Hint: What does the ball of radius $1/2$ around $x$ look like?
-
0I could use some more understanding concerning this question... I would like to keep it going, even if I am not the OP. As for the hint, if x = y then $x^2 + y^2 = 2x^2$ so if we originally had $r^2 = (1/2)^2 = (x^2 + y^2)$ then we would have, by x = y substitution, $1/2 = x\sqrt{2}$. What does this show though? – 2013-09-29