Standard compactness argument I need to know what does "standard compactness argument" means, I google it but But I did not find any satisfactory results.
Any help: Thanks.
 A: For me the canonical compactness argument is the argument that shows that a compact Hausdorff space is regular, in fact showing a common maxim: "compact sets behave like points (often)"
Suppose $x\in X$, $C \subseteq X$ compact and $x \notin C$, all in a Hausdorff space $X$. Then $x$ and $C$ have disjoint open neighbourhoods (just as two distinct points already have).
For every $p \in C$ pick $U_p$ and $V_p$ open in $X$ such that $x \in U_p, p \in V_p , U_p \cap V_p = \emptyset$, which can be done as $x \neq p$ and $X$ is Hausdorff. The $V_p$ cover the compact set $C$, so finitely many cover them, say $C \subseteq V:=V_{p_1} \cup \ldots \cup V_{p_n}$, for finitely many $p_1,\ldots, p_n \in C$. But then $U = \cap_{i=1}^n U_{p_i}$ is also open (a finite intersection!) and contains $x$ and 
$$U \cap V = \cup_{i=1}^n (V_{p_i} \cap U) \subseteq \cup_{i=1}^n (V_{p_i} \cap U_{p_i}) = \emptyset$$ showing that $U$ and $V$ are the required open neighbourhoods of $x$ and $C$.
A totally similar argument, using the above as a lemma, shows that $C$ and $D$ compact disjoint subsets of a Hausdorff space $X$ have disjoint open neighbourhoods as well. 
The compactness allows one to fix things for points, and then using a finite subcover, to fix them for the whole compact set. Many compactness arguments use this idea. 
