Claim made in a proof that a particular space is Lindelof.

[jmjlt88]

I was reading over the proof that the set ℝ with the lower-limit topology is Lindelof. Munkres claimed the Lindelof condition is equivalent to the condition that every open covering of the space by basis elements has a countable subcollection that cover the space. I wanted to write out all the details of the verification of this claim. Basis elements are open sets; hence, one direction is easy. For the other direction, we assume the space X satisfies the condition involving its basis elements and let A be an covering of X by sets opem in X. For each x in X, there is some element of A that contains x; denote it A_x. By the defintion of a basis for a topology, there is some B_x that contains x and lies entirely in A_x.
-----The Question-----
Now, I want to let B be the collection of these basis elements. That is, B is the collection of all basis elements B_x such that B_x contains x and lies entirely in some element A_x of A containing x. Is this construction clear (and legal)? Given an arbitrary basis element B, if for one of its elements x, B lies entirely in some element of A contaning x, then B is a member of B. If this collection is well-defined, then I can finish off the details.

Thank you!

 

[micromass]

Sure, that's the way to do it.

 

[jmjlt88]

Thanks micromass! :)