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

  1. Mar 16, 2013 #1
    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 Ax. By the defintion of a basis for a topology, there is some Bx that contains x and lies entirely in Ax.
    -----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 Bx such that Bx contains x and lies entirely in some element Ax 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!
     
  2. jcsd
  3. Mar 16, 2013 #2

    micromass

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    Sure, that's the way to do it.
     
  4. Mar 17, 2013 #3
    Thanks micromass! :)
     
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