injective images of Baire space

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

Theorem.

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

Although the inverse $f^{-1}:f(𝒩)\to 𝒩$ 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
Canonical name	InjectiveImagesOfBaireSpace
Date of creation	2013-03-22 18:47:12
Last modified on	2013-03-22 18:47:12
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	6
Author	gel (22282)
Entry type	Theorem
Classification	msc 54E50
Related topic	BaireSpaceIsUniversalForPolishSpaces
Related topic	SpacesHomeomorphicToBaireSpace