Closure, Compactness, and Completeness

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Homework Statement



The set S = [0,1] U {3}

Homework Equations




I need to say whether it is closed, open, compact, complete or connected. If it is not compact, give an example why. Same thing for completeness. If its not connected, state why not.

The Attempt at a Solution


I think it is definitely not connected. I also think the set is closed and therefore bounded and compact and complete. But I could be way off...
 
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as always start from the definitions...
 
yes i know...but am i on the right track. i just can't seem to pinpoint whether the set is open or closed. and if i knew that...then i would be fine...
 
ok so start with the definition of open and/or closed...

clearly any neighbourhood of 3 conatins points not in theset, so it cannot be open
 

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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