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

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Every open sub set of Rp 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.
 

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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?
 
This is equivalent to R^p being 2nd-countable.
 
I think I got it. Thank you for your comments
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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