Second-countable implies Lindelof
This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property must also satisfy the second topological space property
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Further information: Second-countable space
Further information: Lindelof space
Given: A second-countable space with countable basis
To prove: If form an open cover of , there exists a countable subcover of among the s
Proof: For each basis element , let be any containing , if such a exists, otherwise, pick nothing. This gives a (at most) countable subcollection of the collection . We want to show that this subcollection covers .
Suppose there exists , that does not belong to any . Then, since the entire collection of cover , we can find some such that . Further, there exists some such that . Now, since is contained in at least one of the s, there should exist an element . But such an element would also contain , contradicting the claim that does not belong to any .
- Topology (2nd edition) by James R. Munkres, More info, Page 191, Theorem 30.3(a), Chapter 4, Section 30