injective images of Baire space


Let X be an uncountable Polish space. Then, there is a one-to-one and continuous functionMathworldPlanetmathPlanetmath f:NX such that Xf(N) is countable.

Although the inversePlanetmathPlanetmathPlanetmath f-1:f(𝒩)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 (

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