@PeterTamaroff Yes, some examples are $\Bbb R$ with basis open sets $[a, b)$, the long line, and $\Bbb R$ with basis open sets of the form $U - \text{countable set}$, where $U$ is open in the standard topology
In Spivak's Different Geometry volume 1 in the appendix we see that requiring a manifold be Hausdorff and metrisisable are equivalent conditions, There are other such conditions.
So yeah screw you @GaloisintheField - as you can see I didn't lie. Don't accuse me of doing that again.
I wouldn't mislead when it comes to something serious @GaloisintheField - you were really offensive.
@GaloisintheField I'm not, I can take a picture of the page in the book if you like.
@GaloisintheField no it doesn't, it says the "manifolds we'll deal with" satisfy that, notice it uses metric ones too, not topologies. If you look in the appendix at the back ....
I haven't got the book to hand, and I don't recall the exact definition of closed (other than it returns to where it starts) so if that isn't it. I'll need the definition and 3 minutes.
I vaguely recall existence of something like "ExerciseWiki", where goal was to collect various mathematical exercises (and solutions). Do I remember it wrong? Or does anybody remembers seeing such thing (and, ideally, has the link)?
maths.kisogo.com/index.php?title=D-system I try and do stuff like this. I found 2 definitions, they're listed, with references and proved equiv. The search is now at the point where it is actually worth using though. Which I am quite proud of.
@Danu I would either go with some space derived from $\omega_1$ or with some weak or weak* topology. Whatever you feel more comfortable working with. You can find some posts on the main with some examples. I listed a few of them here.
