Every open subset of R^p is the union of countable collection of

[vish_maths]

Every open sub set of R^p is the union of countable collection of closed sets.

I am attaching my attempt as an image file. Please guide me on how I should move ahead. Thank you very much for your help.

 

[micromass]

Take an irrational number $a$ in $G$. You can find a certain open ball $B(a,\varepsilon)$ which remains in $G$. What can you tell about the rational numbers in that ball? In particular, if $q\in B(a,\varepsilon)$ is rational, what can you tell about the closed set in the hint?

 

[WWGD]

This is equivalent to R^p being 2nd-countable.

 

[vish_maths]

I think I got it. Thank you for your comments