injective images of Baire space
Every uncountable Polish space is, up to a countable
subset, an injective image of Baire space .
Theorem.
Let be an uncountable Polish space. Then, there is a one-to-one and continuous function such that is countable.
Although the inverse 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 (http://planetmath.org/ProofOfInjectiveImagesOfBaireSpace).