5
$\begingroup$

Consider the space $D := D([0,1], \mathbb{R})$ of real-valued cadlag functions on $[0,1]$. On $D$ one can invent several (more or less) well-known topologies:

  1. the uniform topology $U$: $(D, U)$ is a Banach space which is not separable (thus not Polish)
  2. the Skorokhod $J_1$-topology: $(D, J_1)$ is Polish but not a topological vector space (TVS) since addition is not continuous everywhere
  3. the weaker Skorokhod $M_1$-topology: $(D, M_1)$ is also Polish but again not a TVS (addition is not continuous everywhere but the set of discontinuities is smaller than for $J_1$)
  4. the Skorokhod $J_2$ topology (weaker than $J_1$) and $M_2$ topology (weaker than $M_1$) : $D$ is Lusin (I think not Polish) and not a TVS again
  5. the Jakubowski $S$-topology: $(D,S)$ is Lusin, not metrizable (thus not Polish), addition is sequentially continuous, but not continuous everywhere and thus $(D,S)$ ist not a TVS.

This leads me to the following

Conjecture: On $D$ one can not define a topology such that $D$ is both Polish and a TVS (or even more a separable Banach space).

Does anyone know some attempts to show this or is it an open problem?

$\endgroup$
  • $\begingroup$ Well, technically … $D$ is a real vector space of (algebraic) dimension $2^{\aleph_0}$, as are $\ell^p(\mathbb{N})$, $\mathbb{R}^{\mathbb{N}}$, $C^{\infty}(\mathbb{R})$, and several others. So there are various Polish TVS topologies on $D$. But that may not count as "defining" such a topology on $D$. Is your question about existence or definability? If the latter, which sense of definability? $\endgroup$ – Daniel Fischer Aug 21 '18 at 14:19
  • $\begingroup$ @DanielFischer I mean definability in the sense of being useful for applications and in particular for stochastic processes. But I think, since probability theory basically deals with countable operations or countable constructions (in whatever sense) and convergence of sequences (and not general nets), sequential continuity of addition should be fine (as it is the case for the Jakubowski S-topology mentioned above). Just curious, if there are other applications that require a useful Polish TVS or LCS topoogy on $D$, e.g. for applying theorems for Polish spaces and functional analysis at once. $\endgroup$ – yadaddy Aug 21 '18 at 14:28
  • 1
    $\begingroup$ In that sense, I believe your conjecture holds. A useful topology on $D$ ought to be comparable to the uniform topology, and the open mapping theorem says that a completely metrisable TVS topology on $D$ which is comparable to $U$ must be $U$ itself. $\endgroup$ – Daniel Fischer Aug 21 '18 at 14:36
  • $\begingroup$ @DanielFischer Hmm, yes, right. I admit, the question as stated is rather vague. The usual topologies on the spaces $\ell^p$, $\mathbb{R}^\mathbb{N}$ and so on are in some sense intrinsically defined. So I would also only allow "intrinsic" topologies on $D$ and not consider those that are defined by an external auxiliary bijection to other spaces. $\endgroup$ – yadaddy Aug 21 '18 at 14:43

Your Answer

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

Browse other questions tagged or ask your own question.