1
$\begingroup$

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.
Thank you for your time.

$\endgroup$
2
$\begingroup$
  1. No, it doesn't have to be strictly stronger. Polish spaces are Lusin.

  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.

$\endgroup$
  • $\begingroup$ Thanks a lot! +1 for the point you make it is false that "not Polish implies not Lusin". (Actually, the first thing I thought was that we cannot find a topology stronger than the discrete one, but still in phrasing I got caught on the fly in the fallacy of the converse). $\endgroup$ – Kolmin Oct 6 '16 at 15:40
  • $\begingroup$ I meant the fallacy of denying the antecedent... :) $\endgroup$ – Kolmin Oct 6 '16 at 15:46
  • $\begingroup$ May I ask you one last thing I was thinking about? What is a common space in analysis that is Lusin, but it is not Polish? $\endgroup$ – Kolmin Oct 7 '16 at 7:42
  • 1
    $\begingroup$ @Kolmin: The first example that comes to my mind is the weak topology on a separable Banach space. $\endgroup$ – Nate Eldredge Oct 7 '16 at 12:12
  • 1
    $\begingroup$ @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. $\endgroup$ – Nate Eldredge Oct 7 '16 at 12:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.