Intersection of an infinite set

michonamona
Messages
120
Reaction score
0
Let G_{n} = (-1/n,1/n) for all n in N

let G= \bigcap^{\inft}_{n=1}
 
Physics news on Phys.org
So, what's the question/problem?? The intersection is simply 0 (zero, a single number). Hence the intersection of an infinite number of open sets is not necessarily open, whereas the union is always open, as is the finite intersectiion of them. With the closed sets, it's the other way round :).
 

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...
View full post »

Similar threads

Prove that the intersection of subspaces is compact and closed
Replies
12
Views
2K
Proving intersection of finitely many open sets is open
Replies
1
Views
2K
Prove every subset of countable set is either finite or else countable
Replies
1
Views
2K
Intersection of any 2 intervals --> all intervals intersect
Replies
14
Views
2K
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
Replies
1
Views
1K
Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
Replies
1
Views
1K
Prove ##a + b = b + a## using Peano postulates
Replies
3
Views
1K
Integration over a set in n-dimensional space
Replies
1
Views
1K
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
Replies
3
Views
1K
Prove ##a\cdot b = b \cdot a ##using Peano postulates
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top