Suppose that $X$ is a locally compact topological space. Let $M(X)$ denote the Banach space of regular Borel measures on $X$. It is known that the bidual of $C_0(X)$ is a commutative $C^*-$algebra. Denote the spectrum of $C_0(X)^{**}$ by $\tilde{X}$.

Since $C_0(X)^{**}$ is a unital $C^*-$algebra it can be shown that $C(X^{\infty})$ is a subspace of $C_0(X)^{**}$, where $X^{\infty}$ is the one-point compactification for $X$. Now for each $f\in C_0(X)$ we have that $f(\infty)=0$.

Let $x\in\tilde{X}$ and consider the point mass $\delta_x$. Then $\delta_x|_{C_0(X)}$ is a multiplicative linear functional on $C_0(X)$ and therefore it belongs to $X^{\infty}$. Let $\pi(x)$ be the corresponding element in $X^{\infty}$. Therefore we can define a mapping $\pi:\tilde{X}\to X^{\infty}$.

If $x\in X$ is an isolated point then it can be shown that $\pi^{-1}(x)=\{x\}$. What about the converse, if $\pi^{-1}(x)=\{x\}$ can we conclude that $x$ is an isolated point of $X$?