Every uncountable Polish space is, up to a countable subset, an injective image of Baire space $\mathcal{N}$

Although the inverse $f^{-1}\colon f(\mathcal{N})\rightarrow \mathcal{N}$ will not generally be continuous, it is at least Borel measurable. It can be shown that this is true for all one-to-one and continuous functions between Polish spaces, although here it follows directly from <</A>75#>the construction of $f$ http://planetmath.org/encyclopedia/ProofOfInjectiveImagesOfBaireSpace.html.