Proving Non-metrizable spaces Maybe this question seems easy, but is there a strategy or a property to look for when demostrating a space is non-metrizable? Because a part from the fact that I can't find a metric generating the space, I cannot seem to find a way to prove that such does not exist. 
 A: You could prove that the space fails some sort of separation axiom that metric spaces have (basically any of the $T$'s here, or show it is not first countable.
A: Disprove for a space $X$ a property P such that all metric spaces have property P.
Such P include $X$ is Hausdorff, $X$ is normal, $X$ is perfectly normal, $X$ is first countable.
Also for metric spaces we have that 
being ccc, separable, Lindelöf, second countable are all equivalent (if $X$ is metrisable and has one of these properties, it has all the other ones). If for some $X$ we have that e.g. $X$ is separable but not second countable (as for the Sorgenfrey line  (aka the lower limit topology)) then $X$ cannot be metrisable.
For metric spaces we also have that countably compact, pseudocompact, compact
and sequentially compact are all equivalent, so here the same strategy can be applied. e.g. $\omega_1$ is countably compact but not compact, so cannot be metrisable.
I think most proofs of non-metrisability are done along these lines. 
A totally general method is not really possible I think.
