# Lusin spaces: Issues on the definition, and on spaces used in analysis that are not Lusin

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:

1. 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?]

2. 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.
2. 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.
• @Kolmin: Well, by very definition, any weaker Hausdorff topology on $\Delta(X)$ would accomplish that, but I can't think of any that are commonly used in analysis. Maybe you should sort through what you really want to know and ask a new question. Right now it seems like we are just trying to randomly form connections between things with no clear goal. – Nate Eldredge Oct 7 '16 at 12:39