# Prove that every regular Lindelöf space is normal

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## Homework Statement

So, one needs to prove that every regular Lindelöf space is normal, exactly as the title suggests.

## The Attempt at a Solution

I used the following theorem:

Every regular space with a countable basis is normal.

Now, what we need to prove can be proved very similarily to the proof of the theorem above. It's Theorem 32.1., page 200, in Munkres.

The proof is exactly the same, with one variation.

Let B be a basis for X. We choose a basis element contained in V for every x in A. Now, for any x in X\A, choose a basis element containing X. This collection forms an open cover for X, and since X is Lindelöf, it has a countable subcollection. So, the subcollection of all the basis elements for the elements of A is countable. Hence, the rest of the proof is the same.

I hope it won't be a problem to open Munkres and look at the proof, since it was too long to type, so I decided to be practical.