3 a Closed N c is open b Neither open nor closed Let r Q Then every

3 a closed n c is open b neither open nor closed let

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3. (a) Closed. N c is open. (b) Neither open nor closed. Let r ∈ Q . Then every neighborhood N of r contains points of Q and points of Q . (c) Neither open nor closed. Not closed because 0 is an accumulation point of the set which is not in the set. Not open because for each n every neighborhood of 1 /n contains points which are not in the set. (d) n =1 0 , 1 n = which is both open and closed. (e) { x : x 2 > 0 } = ( -∞ , 0) (0 , ) – open. (f) { x : | x - 2 3 } = [ - 1 , 5] – closed. 4. (a) If u / S , then ( u - , u ) contains a point of S for every > 0 which implies that u S . (b) False. 1 = sup [0 , 1] and 1 [0 , 1]. 5. Let x S and let N ( x, ) be a neighborhood of x . Then N ( x, ) contains a point s 1 S ; the neighborhood of radius / 2 centered at x contains a point s 2 S ; The neighborhood of radius / 3 centered at x contains a point s 3 S ; and so on. 1
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Exercises 1.5 1. (a) True. (b) False. The closed interval [0 , 1] is compact. (c) True. Lemma 14.4 (d) False. The set S = {- 1 } ∪ (0 , 1) ∪ { 2 } has a maximum and a minimum but it is not compact.
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  • Topology, Metric space, Sn, Closed set, General topology, ∩TT ∈T

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