What does it mean by a relatively compact open set? 

This is a basic theorem about topological manifolds but I can't understand some stuff about the proof. What does it mean by "relatively compact open set" in the proof? Also what does it mean by Lindelof Theorem? Regarding Lindelof Theorem, all I can find on the Google is the Phragmen Lindelof Theorem only...
Could anyone please help me?
 A: There is the following "thinning out theorem" (as I like to call it) that is used in the proof:

Suppose $X$ is a topological space with a countable base. Let $\mathcal{B}$ be any base for the topology of $X$. Then there is a countable $\mathcal{B}'$ of $\mathcal{B}$ that is also a base for $X$. (I gave a proof in this thread.)

Then the local compactness shows that all relatively compact open sets (open sets with a compact closure) are a base for the topology (because such sets are local bases at each point, by the local Euclideanness, as shown in the beginning of the proof), and so the above theorem gives us a countable local base of open sets with compact closure. It follows that $M$ is a countable union of compact sets $\overline{V_i}$. I think the "Lindelöf theorem" then refers to the fact that a Lindelöf (every open cover has a countable subcover) regular space is normal, although Kelly in his book [cited in your quote]) calls the fact that a space with a countable base is Lindelöf the "Lindelöf theorem". So $X$ is Lindelöf in two ways: first because of the countable base, second from $\sigma$-compactness. Urysohn does not need normality (Kelly formulates it as "a regular $T_1$ space with a countable base is metrisable"), and regular already follows from local compactness together with Hausdorffness.
So if $X$ a manifold means Hausdorff, second countable and locally homeomorphic to some $\mathbb{R}^n$, the latter gives us local compactness and hence regular.
Then Urysohn immediately tells us that $X$ is metrisable; $X$ normal and $\sigma$_compact are then almost trivial consequences; only the $\sigma$-compactness needs the thinning out theorem or Kelly's version of Lindelöf's theorem (reducing the cover of relatively compact open sets to a countable one).
