Question

There is a Haussdorff Hausdorff gap, a sequence $\{f_\alpha,g_\alpha:\alpha\lt\omega_1\}$ of $\omega\to\omega$ functions such that $f_\alpha\lt^*f_\beta<^*g_\beta<^*g_\alpha$ hold for $\alpha\lt\beta\lt\omega_1$ (here $f\lt^* g$ denotes eventual dominance, i.e., that $f(n)\lt g(n)$ holds for large $n\lt\omega$) and there is no function $f:\omega\to\omega$ such that $f_\alpha\lt^* f\lt^* g_\alpha$ holds for $\alpha\lt\omega_1$. If we identify the reals with ${}^\omega\omega$ and set $H_\alpha=\{f:f_\alpha\lt^* f \lt^* g_\alpha\}$ then $\{H_\alpha:\alpha\lt\omega_1\}$ is a decreasing sequence of nonempty $F_\sigma$ with empty intersection.