I have a question in $\beta\mathbb{R}$, the Stone-Cech compactification of the real line $\mathbb{R}$. My question is: is $\beta\mathbb{R}$ a $\mathrm{F}$-space (https://en.wikipedia.org/wiki/F-space) or a Frechet space? I know that $\beta\mathbb{R}\setminus\mathbb{R}$ is a $\mathrm{F}$-space, but not if the whole space has this property.

Thank you for your help in advance :)

I have a question in $\beta\mathbb{R}$, the Stone-Cech compactification of the real line $\mathbb{R}$. My question is: is $\beta\mathbb{R}$ a $\mathrm{F}$-space, i.e., the closure of two disjoint open $F_{\sigma}$-sets are disjoint? I know that $\beta\mathbb{R}\setminus\mathbb{R}$ is a $\mathrm{F}$-space, but not if the whole space has this property.

Thank you for your help in advance :)