Hausdorff spaces from filters I'm sure I'm just being silly, but I've run into a claim in a paper I'm reading which I don't understand.
Suppose $\mathcal{F}$ is a filter on $\mathbb{N}$. There is a natural topology $\tau_\mathcal{F}$ on $\mathbb{N}$ associated to $\mathcal{F}$ - namely, just take as opens exactly the sets in $\mathcal{F}$, plus the empty set. Of course, $\tau_\mathcal{F}$ is never Hausdorff (I require filters to not contain $\emptyset$).
However, on the second page of Todorcevic/Uzcategui's paper Analytic topologies on countable sets there is the following claim:

given a filter $\mathcal{F}$ over $\mathbb{N}$ by an elementary construction it is easy to define a Hausdorff topology of the same complexity as the filter $\mathcal{F}$.

They do not, however, give a reference, and my google-fu fails me due to the sheer number of unrelated hits for queries like "Hausdorff filter." My question is simply, what construction do they have in mind? 
I've included the "descriptive-set-theory" tag both because the paper in question involves descriptive set theory and because I'm interested in the "same complexity" aspect of the construction (which is a descriptive set theoretic concern) as well as its Hausdorffitude.
 A: I think you only refer to the intro of that paper, where the authors, in my opinion, declare what they are going to discuss later in the paper in more detail. In particular, in the same paper, Example 2.3 they give "an elementary method to construct a Hausdorff topology based on a filter", as stated just before the example. I didn't attempt to read the details or understand the construction (but it apears they may have more than one possible construction in mind, and illustrate some of them in their paper, in some detail). 
To make this answer a bit more self-contained I copy Example 2.3 from their paper here. 
Example 2.3. Let $\mathcal F$ be  a  filter  over $\omega$.  We  define  a  topology $\tau(\mathcal F)$ over $\omega+1$ by 
$\tau(\mathcal F)=\{\{\omega\}\cup A: A\in\mathcal F\}\cup\mathcal P(\omega)$. It  is  clear  that  if $\mathcal F$ is  non-principal then $\tau(\mathcal F)$ is a  Hausdorff topology. Since  the  function $f:2^\omega\to 2^{\omega+1}$ given  by $f(A)=A\cup\{\omega\}$ is continuous and $A\in\mathcal F$ iff $f(A)\in\tau(\mathcal F)$, then $\mathcal F$ is Wadge reducible to $\tau(\mathcal F)$. Also notice that if $\mathcal F$ is a non-trivial filter, then $\omega$ is the only limit point of $(\omega+1,\tau(\mathcal F))$. In fact, it is clear that this is a characterization of such spaces. (etc.) 
I tried to read a bit more, Example 5.1 gives another construction of a topology from a filter, this time $T_2$, zero-dimensional. 
Example  5.1. Let $\mathcal F$ be  a  filter  over $\Bbb N$ containing  the  filter  of  cofinite  sets.  Define  a topology over $X=\omega^{<\omega}$ as follows: $U\in \tau_{\mathcal F}\iff \{n\in\Bbb N : s\hat{} n\in U\}\in \mathcal F$ for all $s\in U$. It is clear that $\tau_{\mathcal F}$ is $T_2$, zero-dimensional and has no isolated points. From the definition of $\tau_{\mathcal F}$ is easy to check that $\tau_{\mathcal F}$ is $\Pi_{\alpha+1}^0$ if $\mathcal F$ is $\Pi_{\alpha+1}^0$ or $\Sigma_\alpha^0$. 
Further down:
Of special interest is the case of $\tau_{\mathcal F}$ when $\mathcal F$ is the filter of cofinite sets which we are going to denote simply by 
$\tau_{\mathrm{FIN}}$. We will show that $\tau_{\mathrm{FIN}}$ does not admit an $F_\sigma$ base (the same argument applies to $\tau_{\mathcal F}$ for any free filter $\mathcal F$).
