Hausdorff spaces from filters

Solution 1:

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 $\mathcal F$).