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 April 15th, 2018, 01:12 PM #1 Newbie   Joined: Jul 2014 From: Wrexham Posts: 20 Thanks: 0 Open cover I’m using the usual Euclidean metric on ℝ^2 and the induced metric on P. Am I correct in thinking that a dp-open cover of P={(x, cosx) x∈ℝ} would be {(x, cosx) x∈(-n,n):n∈ℕ}? Also is {ℝ^2} be a d-open cover that is finite and {(x,y)∈(-n,n) y∈ℝ} be a d-open cover that is not finite?
April 17th, 2018, 08:51 AM   #2
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Quote:
 Originally Posted by AJ235 I’m using the usual Euclidean metric on ℝ^2 and the induced metric on P. Am I correct in thinking that a dp-open cover of P={(x, cosx) x∈ℝ} would be {(x, cosx) x∈(-n,n):n∈ℕ}?
The way you have written this, no, it is not a cover, it is a collection of individual points, not open sets. It does include all real numbers but not sets of real numbers.

Quote:
 Also is {ℝ^2} be a d-open cover that is finite and {(x,y)∈(-n,n) y∈ℝ} be a d-open cover that is not finite?
Yes, R^2 is the set of all pairs of real numbers so that single set covers all of R^2 and so any subset of it. The other, {(x,y)∈(-n,n) y∈ℝ}, is again a collection of individual points, not a collection of sets.

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