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6 corrected spelling of "Hausdorff"

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.

5 Replaced < with \lt

There is a Haussdorff gap, a sequence $\{f_\alpha,g_\alpha:\alpha<\omega_1\}$\{f_\alpha,g_\alpha:\alpha\lt\omega_1\}$ of$\omega\to\omega$functions such that$f_\alpha<^*f_\beta<^*g_\beta<^*g_\alpha$f_\alpha\lt^*f_\beta<^*g_\beta<^*g_\alpha$ hold for $\alpha<\beta<\omega_1$ \alpha\lt\beta\lt\omega_1$(here$f<^* f\lt^* g$denotes eventual dominance, i.e., that$f(n)$H_\alpha=\{f:f_\alpha<^* f(n)\lt g(n)$ 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 $\{H_\alpha:\alpha<\omega_1\}$\{H_\alpha:\alpha\lt\omega_1\}$ is a decreasing sequence of nonempty$F_\sigma$with empty intersection. 4 Rollback to Revision 1 There is a Haussdorff gap, a sequence $\{f_\alpha,g_\alpha:\alpha<\omega_1\}$ of$\omega\to\omega$functions such that$f_\alpha<^*f_\beta<^*g_\beta<^*g_\alpha$hold for$\alpha<\beta<\omega_1$(here$f<^* g$denotes eventual dominance, i.e., that$f(n)

If we identify the reals with ${}^\omega\omega$ and set $H_\alpha=\{f:f_\alphaf(n)$H_\alpha=\{f:f_\alpha<^* f <^* g_\alpha\}$then$\{H_\alpha:\alpha<\omega_1\}$is a decreasing sequence of nonempty$F_\sigma\$ sets with empty intersection.

 THE PROGRAM LEAVES OUT HALF OF MY ANSWER! 
 
 
 
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