Definition:Uncountable Discrete Ordinal Space

Definition
Let $\Omega$ denote the first uncountable ordinal.

Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$.

Let $S$ be the set of points of $\hointr 0 \Omega$ of the form $\alpha + 1$, where $\alpha$ is a limit ordinal.

Let $\tau$ be the subspace topology induced by the order topology on $\hointr 0 \Omega$.

$\struct {S, \tau}$ is known as the uncountable discrete ordinal space.