If a topological space $X$ is locally compact and I have that $x \in C$ for $C$ compact, is there a $U$ of $x$ such that $x \in U \subset C$

Question

The definition of a locally compact topological space $X$, according to my notes, is that for every $x \in X$, there exists compact $C \subset X$ such that $x \in C$ with a neighbourhood $U$ of $x$ with $x \in U \subset C$.

My question is a slight variation of this. If a topological space $X$ is locally compact and I have that $x \in C$ for $C$ compact, does there exist a neighbourhood $U$ of $x$ such that $x \in U \subset C$. The difference is now I am starting with a compact set.

Answer 1

Take $X = \mathbb{R}$, $x \in X$ and $\{x\} = C$.

Answer 2

But $C$ might have no interior.

For example, let $X=\mathbb{R}^2$, let $x$ be any point of $X$, and let $C$ be a line segment through $x$.

Even if $C$ has interior, the point $x$ might be on the boundary of $C$.

For example, using $X=\mathbb{R}^2$ again, let $x$ be any point of $X$, and let $C$ be a closed disk whose circular boundary passes through $x$.