TITLE: Show that the topological space ( X, $\tau$ ) is not metrizable
QUESTION [7 upvotes]: For the topological space ( X, $\tau$ ), with X = {0, 1} and $\tau$ = { $\emptyset$ , {0}, {0,1} } , prove that ( X, $\tau$ ) is not metrizable. 
I know intuitively it can't be but don't know how to prove it. The only idea I have is that {0, 1} cannot be open as X = {0, 1} is discrete but I'm not even sure if this is right.

REPLY [2 votes]: An easy proof uses the following: A finite topological space is $T1$ (and anything stronger, like metrizable) iff its topology is the discrete topology.
Assuming this, we see that $\tau = \{ X , \emptyset, \{0\}\}$ is not metrizable because it's not the discrete topology (it's missing $\{1\}$ as an open set).