TITLE: Why a point and compact set being separated doesn't imply point being separated from every point of that compact set?
QUESTION [0 upvotes]: Recently I've started studying dimension (inductive) theory and there's a logic behind the statement below I don't understand, so I need help. If there already exists a duplicate to my question, I apologize in advance, but I haven't found any. I'm only hoping answer isn't trivial.
Let $\left(X,\mathcal{T}\right)$ be separable, metrizable topological space. 
Statement   Let $p \in X$ and $C$ nonempty, compact subset of $X$ not containing $p$. If $p$ and $c$ are separated in $X$ for every $c \in C$, then $p$ and $C$ are also separated in $X$.
Problem I don't understand why converse isn't true and why is $C$ being compact important.
Definition I'm using is:
Disjoint sets $A,B \subset X$ are said to be separated in $X$ if there exists a clopen set $U \subset X$ such that $A \subset U$ and $B \subset X/U $.

REPLY [1 votes]: Importance of compactness: Let $p=0$, $C=\{1,1/2,1/3,\ldots\}$, $X=\{p\}\cup C$.
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