# injective images of Baire space

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

###### Theorem.

Let $X$ be an uncountable Polish space. Then, there is a one-to-one and continuous function $f\colon\mathcal{N}\rightarrow X$ such that $X\setminus f(\mathcal{N})$ is countable.

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 the construction of $f$ (http://planetmath.org/ProofOfInjectiveImagesOfBaireSpace).

Title injective images of Baire space InjectiveImagesOfBaireSpace 2013-03-22 18:47:12 2013-03-22 18:47:12 gel (22282) gel (22282) 6 gel (22282) Theorem msc 54E50 BaireSpaceIsUniversalForPolishSpaces SpacesHomeomorphicToBaireSpace