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Homework Statement
This question has two parts :
(a) Let F be a finite collection of closed sets in ℝn. Prove that UF ( The union of the sets in F ) is always a closed set in ℝn.
(b) Let F be a finite collection of closed intervals An = [[itex]\frac{1}{n}, 1- \frac{1}{n}[/itex]] in ℝ for n = 1,2,3... What do you notice about UF? Is it closed, open, both or niether?
Homework Equations
I know a set S is open ⇔ it is equal to its own interior, that is S = S0.
I know a set S is closed if its compliment ℝn-S is open.
I also know a set S is closed ⇔ B(S) [itex]\subseteq[/itex] S.
I know that the only open and closed sets are ℝn and ∅.
I know a set S is neither open or closed if there is at least one point which has a neighborhood containing points from both S and its compliment.
The Attempt at a Solution
(a) Suppose that F = {[itex] S_1, S_2, ..., S_p | S_i\in ℝ^n, 1 ≤ i ≤ p [/itex]}
Suppose further that :
[tex]Q = \bigcup_{i=1}^{p} S_i[/tex]
We want to show Q is always a closed set in ℝn, so let us show that the compliment of Q, ℝn - Q is open.
We know from F that : [itex]S_i \subseteq B(S_i), \space 1≤ i ≤ p[/itex]. That is, each individual set contains its own boundary.
This implicates to us that : [itex]ℝ^n - S_i[/itex] is open since Si is closed for 1 ≤ i ≤ p now suppose that :
[tex]Q_o = \bigcup_{i=1}^{p} (ℝ^n - S_i)[/tex]
So that Qo is composed of a finite collection of the compliments of F.
Am I on the right track here or have I missed something? Ill attempt (b) after I figure this one out.