There are many minor variations of this definition in use: the T1 condition can be replaced by T2 or T3, and some authors allow a countable or even arbitrary number of isolated points.
The existence of a Luzin space is independent of the axioms of ZFC.
Lusin (1914) showed that the continuum hypothesis implies that a Luzin space exists.
Kunen (1977) showed that assuming Martin's axiom and the negation of the continuum hypothesis, there are no Hausdorff Luzin spaces.
By the continuum hypothesis, it is possible to enumerate them as Sα for countable ordinals α.