Minimal Hausdorff topology S98 needs elaboration

[felixpernegger]

1. It's unclear to me what the points in the def. of A actually stand for
2. What are ideal points?
3. No proofs (all copied from counterexamples book)

[Moniker1998]

I believe this is a typical construction of the type "every point is isolated except for those points" and the authors decided to call them ideal points, for some reason. I don't like how this is written but I think I understand it.

The points of $A$ should be $\omega, \omega^\ast$ (the dual of $\omega$) and $\omega$ itself as a point between them, made so that $x\in \omega, y\in \omega^\ast$ then $x < y$ (so that elements of $\omega^\ast$ are denoted as negative numbers).

This reminds me of a lot of similar constructions when it comes to semi-regular spaces and what not.
(minimal Hausdorff spaces are precisely semiregular $H$-closed spaces, so this is clearly related)