Metric Space and Lindelof Theorem

[blackangus]

Homework Statement

Assume some metric space (K,d) obeys Lindelof, take (X,d) a metric subspace of (K,d) and show it too must obey Lindelof.

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 {J_i | 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 {J_i} 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!

 

[micromass]

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]

No we haven't learned about second countable yet, so I shouldn't be using that in my proof. Any other suggestions? Thank you!