I have this question for several days:
Let $X$ be a topological space and $X$ is Hausdorff. $C$ is an countable discrete subset of $X$. Then does there exist a disjoint family of open sets $\{U_x: x\in C\}$ of $X$ such that $x\in U_x$ for all $x\in C$?
My idea is this: Assume that $C= \{x_n:n\in N\}$. Induction on $N$. For $n=1$, for $C$ is discrete, we choose an open set $U_1$ such that only $x_1 \in U_1$; now we assume for n=k, we have got $\{U_i: i=1,2,...,k\}$ such that they are disjoint. When $n=k+1$, for $x_{n+1}$, because $X$ is Hausdorff, so I can choose an open set $U_{n+1}$ which disjoint with any $U_i$ for any $i=1,2,...k$. Therefore I can get such disjoint family.
Am I right? Thanks for any help.