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As is well-known, a continuous image of a separable and completely metrizable space is called in Topology as analytic set or analytic space. The following theorems Baire-Alexandroff Theorem (1929): A topological space $X$ is homeomorphic to $\mathbb{P}$ iff X is a separable, 0-dimensional and metrizable space which is an absolute $G_\delta$ having no compact open subset. Hausdorff Theorem (1932): A metric space is an analytic set iff it is a continuous image In the above setting, $\mathbb{P}$ denotes the irrationals with the standard subspace topology obtained . We will give a detailed proof in this talk of the following well-known theorem of Ernest Michael and Arthur H. Stone from 1969: The Main Theorem: Every analytic set is actually an image of $\mathbb{P}$ under a quotient mapping.