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By Engler D. A.

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13) > 0 such that (x − , x + ) ⊂ U. 14) Thus x ∈ / U , so x ∈ C and C is closed. D. 4. For n = 1, 2, 3, . , let In = − n1 , n+1 . Is n ∞ In n=1 open or closed? 5. For n = 3, 4, 5, . , let In = 1 n−1 n, n . Is ∞ In n=3 open or closed? 6. Suppose, for n = 1, 2, 3, . , the intervals In = [an , bn ] are such that In+1 ⊂ In . If a = sup{an : n ∈ Z+ } and b = inf{bn : n ∈ Z+ }, show that ∞ In = [a, b]. 3. 7. Find a sequence In , n = 1, 2, 3, . , of closed intervals such that In+1 ⊂ In for n = 1, 2, 3, .

And ∞ In = ∅. 9. Suppose Ai ⊂ R, i = 1, 2, . . , n, and let B = that n i=1 Ai . Show n B= Ai . 10. Suppose Ai ⊂ R, i ∈ Z+ , and let ∞ B= Ai . i=1 Show that ∞ Ai ⊂ B. i=1 Find an example for which ∞ B= Ai . 11. Suppose U ⊂ R is a nonempty open set. For each x ∈ U , let Jx = where the union is taken over all (x − , x + δ), > 0 and δ > 0 such that (x − , x + δ) ⊂ U . a. Show that for every x, y ∈ U , either Jx ∩ Jy = ∅ or Jx = Jy . b. Show that U= Jx , x∈B where B ⊂ U is either finite or countable.

Since Cα is closed, it follows that x ∈ Cα for every α ∈ A. Thus x ∈ α∈A Cα and α∈A Cα is closed. D. 4. Suppose C1 , C2 , . . , Cn is a finite collection of closed sets. 9) i=1 is closed. n Proof. Suppose {ak }k∈K is a convergent sequence with ak ∈ i=1 Ci for every k ∈ K. Let L = lim ak . Since K is an infinite set, there must exist an integer k→∞ m and a subsequence {anj }∞ j=1 such that anj ∈ Cm for j = 1, 2, . .. Since every subsequence of {ak }k∈K converges to L, {anj }∞ j=1 must converge to L.

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