According to Wikipedia, Baire space is defined as "the Cartesian product of countably infinite copies of the natural numbers". The page states later on that Baire space is homeomorphic to the set of irrational numbers, which seems to indicate that it is uncountable; also, its definition would seem to make it equivalent to the set of infinite sequences of natural numbers, which should also be uncountable by Cantor's diagonal argument.(adsbygoogle = window.adsbygoogle || []).push({});

However, in the sequence of transfinite ordinals, there is the ordinal ##\omega^{\omega}## (the same symbol is used for Baire space according to Wikipedia), which is clearly stated to be countable. Yet this ordinal would seem to be equivalent to Baire space. Is it? If so, is Baire space actually countable? How could it be, given the above?

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# Is Baire Space countable?

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