Proof That Every Lindelof Metric Space is Separable

Theorem

Every lindelof metric spaceis separable.

Proof

Suppose the metric spaceis Lindelof. Letbe a real number and letbe a maximal subset ofsuch thatfor every(such a maximal subset is guaranteed by Zorn's Lemma).

For eachconsiderandis open andis an open covering ofby open sets. Sinceis Lindelof, a countable subcover exists.

Suppose we removefrom the covering for someThe remaining sets would not coversince non of them would containHenceis countable, andis countable.

We can repeat this construction for eachWe obtain corresponding maximal subsets

Letis countable as the union of countably many countable sets.

SupposeandSetThen there issuch thatLetbe any nonempty open subset ofChooseandsuch that thencontains an element ofandis dense inhenceis separable.