Moniker1998
Moniker1998
@yhx-12243 oh sorry I forgot that existed, since it was sitting there for so long. Not quite, but it's similar. And also showing that $\beta\mathbb{N}$ is non-normal. Moreover it's more...
As in #1044 those spaces are LOTS. This makes me think that perhaps every LOTS is cozero complemented.
This is already known, in *Infinite-dimensional topology of function spaces* by van Mill it exists as exercise A.12.10. I don't really know a good reference, but its probably there somewhere....
@felixpernegger thanks Speaking of, https://topology.pi-base.org/spaces/S000115/properties/P000089 It's unknown if extended topologist sine curve has this property
@prabau such spaces are not called basically disconnected. That's a different property
@prabau Yes. Basically disconnected implies every closure of a cozero set is a zero set, which implies that the space is cozero complemented. It'd be nice to add either the...
@prabau that's right, you can easily prove 3) from 1) and 2). I find it easier to say that $U\cap D$ is open in $D$, and $D$ is dense in...
> And like you said, it would be good to have the remainder β N ∖ N as an example in pi-base, to show extremally disconnected is not always preserved...
We might as well add first measurable cardinal with discrete topology.
[Another space of this type](https://math.stackexchange.com/a/5022661/476484), which is essentially strong uniform topology on $\nu(\kappa)$ above.