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).

单词	InjectiveImagesOfBaireSpace
释义	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).
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