I have two questions concerning Lusin spaces. Before the questions, the definition:
Definition (Lusin Space): Given a Hausdorff space $(X, \tau)$, that space is said to be Lusin if there is a stronger topology $\tau’$ on $X$ such that $(X , \tau’)$ is Polish.
Below there are my questions, where – in italics – there are my thoughts over them.
Questions:
Has the topology $\tau’$ to be strictly stronger than $\tau$?
[I would say not. For example, $\mathbb{R}$ endowed with the euclidean topology is Hausdorff, and it is polish, thus with that topology it is already Lusin, right?]In the book, I found that most of the spaces encountered in analysis are Lusin. What are those spaces that are not Lusin?
[I can think of a space that is not Lusin, but I don’t think it is a workhorse in analysis… Take a uncountably infinite space $(X, \tau_d)$ endowed with the discrete topology: then it is metrizable, and complete, but not separable, because it is not second countable, hence not polish, and from point 1, not Lusin, right?]
As always, any feedback is most welcome.
Thank you for your time.
1 Answer
$\begingroup$No, it doesn't have to be strictly stronger. Polish spaces are Lusin.
Most spaces encountered in analysis are Polish. You are right that an uncountable discrete space is a counterexample (but be a little careful: "not Polish" does not imply "not Lusin". The key here is that there are no other topologies stronger than the discrete topology.) Any other non-separable metric space works too; $\ell^\infty$ is another example.
