A measurable space ($\Omega$,$\mathcal F$) is called a standard measurable space if it is Borel isomorphic to one of the following measurable spaces: $(\langle 1, n\rangle,\mathcal B(\langle 1, n\rangle))$, $(\mathbb N,\mathcal B(\mathbb N))$ or $(M,\mathcal B(M))$, where $\langle 1, n\rangle = \{1, 2, \ldots , n\}$ with the discrete topology, $\mathbb N = \{1, 2, \ldots \}$ with the discrete topology and $M=\{0,1\}^{\mathbb N}$ with the product topology.
Where Borel isomorphic means there exists a bijection which is a measurable mapping.
My question is as follows: Why is a Polish space a standard measurable space??