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?