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Metric Space and Lindelof Theorem

by blackangus
Tags: lindelof, metric, space, theorem
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blackangus
#1
Oct31-11, 06:20 PM
P: 2
1. The problem statement, all variables and given/known data
Assume some metric space (K,d) obeys Lindelof, take (X,d) a metric subspace of (K,d) and show it too must obey Lindelof.

3. The attempt at a solution
I'm assuming since I know that (K,d) obeys Lindelof then there is some open cover that has a countable subcover say {Ji | i is a positive rational number}. Since this covers all of the metric space, then it surely covers the metric subspace. So I'm thinking all I need to really prove is that the elements of this set of open countable covers is open under metric subspace (X,d)? Or do I need to show that I can select a countable number of {Ji} to cover (X,d)? If so, any suggestions on how to do so?

This is the last assignment question, and it's due this Wednesday November 2nd, any help would be appreciated!
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micromass
#2
Oct31-11, 08:40 PM
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P: 18,248
It will be hard (even impossible) to prove this directly. For the simply reason that the result isn't true in an arbitrary topological space.

You need to do a little shortcut. Do you know that in a metric space that Lindelof is equivalent to being second countable?? (prove it if you don't know)
Well, using this, it suffices to prove that a subspace of a second countable space is second countable. This is a lot easier.
blackangus
#3
Nov1-11, 06:08 AM
P: 2
No we haven't learned about second countable yet, so I shouldn't be using that in my proof. Any other suggestions? Thank you!


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