Secondcountable 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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Contents
Statement
Propertytheoretic statement
The property of topological spaces of being a secondcountable space implies, or is stronger than, the property of being a Lindelof space.
Verbal statement
Any secondcountable space is Lindelof.
Definitions used
Secondcountable space
Further information: Secondcountable space
A topological space is termed secondcountable if it admits a countable basis.
Lindelof space
Further information: Lindelof space
A topological space is termed Lindelof if every open cover of the space has a countable subcover.
Proof
Given: A secondcountable 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 .
References
Textbook references
 Topology (2nd edition) by James R. Munkres, ^{More info}, Page 191, Theorem 30.3(a), Chapter 4, Section 30