# Covering number of the meager ideal

I learnt from a paper that "Let cov(K) be the least cardinal k such that a perfect Polish space can be expressed as a union of k meager sets. (It does not matter which perfect Polish space is used to define cov(K) ) " . I don't know why cov(K)'s are equal for different perfect Polish spaces. I want to use this result in my own paper, but I fail to find any books contain this result, could anybody kindly to help me ?

• Not points' but nowhere dense sets' I think:after you remove the boundaries of the memebrs of some countable base what is left is zero-dimenional and Polish and perfect and nowhere locally compact, hence homeomorphic to the space of irrationals. NB the plane minus a countable set and the line minus a countable set are not homeomorphic: the former is connectedm the latter is not. – KP Hart Sep 15 '10 at 17:33
• Oh, and to avoid confusion: Baire space' (capitalized) is the topological product N^N, whereas a baire space' is a space that is not meager in itelf, It seems to me that you confused the two but I might be wrong. – KP Hart Sep 16 '10 at 6:58
Andreas gave an argument while I was typing, so let me give a different approach: The proof that the covering number is the least $\kappa$ for which ${\sf MA}_\kappa({\rm countable})$ fails is soft enough that it can be easily adapted to work for any Polish space you begin with. Details of this equivalence are, for example, in the excellent Set theory: on the structure of the real line'' by Tomek Bartoszynski and Haim Judah, that you will want to have as a reference anyway.