Why is indiscrete topology unmetrizable? > For instance, the indiscrete topology for $X$ cannot arise from a metric when $X$ has more than one point. One way to see this is to note that the complement of a one-point set in a metric space is always open. I don't see what is going on here. Say $X=\\{a,b\\}$. Then the indiscrete topology for $X$ would be $\mathfrak{F}=\\{\emptyset,X \\}$. But now what?

What you need to do now is assume there exists a metric $d$, and derive a contradiction. For instance, if $X = \\{a,b\\}$, then there is some distance $D = d(a,b) > 0$. What can you now say about the (open) ball of radius $D$ centered at $a$? In other words, which points $x\in X$ satisfy $d(a,x) < D = d(a,b)$?