Clarification on the Hausdorff property If $X$ is a Hausdorff space then for points $a,b \in X$ there are disjoint open sets $U$ and $V$ such that $a \in U$ and $b \in V$.  So, take a set of points $\{a_1, \ldots , a_n\}$ and another point $x$.  Then for each $a_i$ there are disjoint open sets $U_i$ and $V$ containing $a_i$ and $x$, respectively.
My question is: is $V$ disjoint from all the $U_i$, or does this only hold pairwise?  Are there conditions we can impose to guarantee an open set separating $x$ from all points in some given finite set?
 A: Let $\{a_1, ..., a_n\}$ be a finite collection of points. Let $x$ be a point different from all the $a_i$'s. For each $i$, by being Hausdorff, there exists $U_i$ and $V_i$ such that $a_i \in U_i$, $x \in V_i$ and $U_i \cap V_i = \emptyset$. 
Then $V = \bigcap_{1 \leq i \leq n} V_i$ is an open set (being a finite intersection of open set) containing $x$ which is disjoint from all the $A_i$'s. 
A: I just want to point out that with the idea in William's proof, above, you can easily prove the following (seemingly more general) property:
If $X$ is Hausdorff and $x_1, \ldots , x_n \in X$ are distinct, then there are open neighbourhoods $V_1 , \ldots , V_n$ about $x_1, \ldots , x_n$, respectively, such that $V_i \cap V_j = \emptyset$ for $i \neq j$.
A: William's answer shows that for your example, the set $A=\{a_1,\ldots,a_n\}$ can be separated from the one-point set $B=\{x\}$ in any Hausdorff space.  ("Separated" here means there are disjoint open sets $U$ and $V$, containing $A$ and $B$ respectively.)  But as you can see, the proof only goes through because $A$ is finite.
A more general property, which holds for infinite sets $A$, is called "regularity".  A regular space is one in which every closed subset $S$ can be separated (in the sense of the previous paragraph) from each single point $x$.  If single points are closed, then this implies that the space is also Hausdorff, by taking $S$ to consist of some single point $y$. Hausdorff spaces may fail to be regular, however.
There is a whole family of these so-called separation axioms, and study of the relations between them is a significant field of study.
