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April 15th, 2018, 12:06 PM   #1
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Open covers

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?
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April 15th, 2018, 12:36 PM   #2
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What is a dp open cover? A d open cover? Please define any terms not in general use. The only meaning of dp I know is not suitable for a family-oriented website such as this.

Quote:
Originally Posted by AJ235 View Post

{(x, cosx) x∈(-n,n):n∈ℕ}?
That's a set of isolated points in the plane. It's not an open set nor a collection of open sets. It couldn't be an open cover of anything.

Last edited by Maschke; April 15th, 2018 at 12:38 PM.
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April 15th, 2018, 12:45 PM   #3
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not that dp! The d part is emphasising that I'm using the Euclidean metric and the dp is emphasising I'm using the metric induced on P

Last edited by AJ235; April 15th, 2018 at 01:01 PM.
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April 15th, 2018, 01:25 PM   #4
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This is what I was given as an example. The blue is the dp-open cover.
Attached Images
File Type: jpg D24CA7AC-8744-45FC-AFF3-C7A9462CDA11.jpg (19.9 KB, 14 views)
File Type: jpg 8BEF4E6D-4778-406A-9064-4636E2D14797.jpg (25.0 KB, 7 views)

Last edited by AJ235; April 15th, 2018 at 01:33 PM.
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April 17th, 2018, 07:57 AM   #5
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As I said in your other post, under "Topology", a 'cover' has to be a collection of sets, not individual points so, no, this is not a cover for the set.

I suspect that you intended to say {{(x, x^2)}: x∈-n, n), n∈ N}, each member is the set of all such pairs, not individual pairs.

Last edited by Country Boy; April 17th, 2018 at 08:11 AM.
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