TITLE: What is Baire's zero-dimensional metric space?
QUESTION [3 upvotes]: I'm not familar with metrizable spaces. I met a notation: Baire's zero-dimensional metric space. Could somebody explain it for me? Thanks ahead:)

REPLY [5 votes]: The term almost certainly refers to the product space $\omega^\omega$, the product of countably infinitely many copies of the countably infinite discrete space, which is well-known to be homeomorphic to the space of irrational numbers with the usual topology. More generally, if $D_\kappa$ is the discrete space of cardinality $\kappa$, $D_\kappa^\omega$ is often referred to as the zero-dimensional Baire space of weight $\kappa$.
Added: Having now seen your question here, I’ll add that one can of course replace $D_\kappa$ with any set that has been given the discrete topology, as has been done in that paper that you cite there: in that paper $N(A)$ is the space $A^\omega$, provided that $A$ is given the discrete topology. It is therefore homeomorphic to $D_\kappa^\omega$, where $\kappa=|A|$. All of these spaces are metrizable; I generally define the metric by setting $d(x,y)=2^{-n}$, where $n=\min\{k\in\omega:x_k\ne y_k\}$, if $x\ne y$, and of course $d(x,x)=0$.
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