In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if $g:Y\to X$ is a quasi-homeomorphism, $Z$ a sober space and $f:Y\to Z$ a continuous map, then there exists a unique continuous map $F:X\to Z$ such that $F\circ g=f$. Let $X$ be a $T_0$-space, $q:X\to {}^{s}X$ the injection of $X$ onto its sobrification ${}^{s}X$. It is shown, here, that $q\left(\text{Gold}\left(X\right)\right)=\text{Gold}\left({}^{s}X\right)$, where $\text{Gold}\left(X\right)$ is the set of all locally closed points of $X$. Some applications are also indicated. The Jacobson prime spectrum...