I have started reading about Baire Space as a prerequisity for the subject of Borel hierarchies. And if I understand correctly, Baire space is $\omega^\omega$ (The space of all functions from $\omega$ to $\omega$)? I guess the Topology on this space is the product topology? also, Can I assume that the cardinality of $\omega^\omega$ is $\aleph_0$ as a countable union of countable sets $\bigcup_{k=0}^\infty \omega^k$?.
If the answer is yes, Then, what is the first ordinal that has cardinality $\aleph_1$. Also, is anyone familiar with a good source for studying Borel hierarchy (That also contain exercises if possible..)?
Thanks!! Shir
Now that I think of it, Baire space should also contain uncountable sequences.. so, what is the cardinality of $\omega^\omega$? is it $\aleph_0$ or $\aleph_1$?