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  "body": "\n## Description\n\nThe *Baire space* $\\N^{\\N}$ is the set of all infinite sequences of [[natural numbers|0MWKJ44V5A]].\nIts topology is the [[product topology|0GBG8B49AA]] of countably many copies of the discrete space $\\N$.\nIts basic open sets are the sets of all sequences that begin with a given finite sequence.\nThe topology comes from the metric $d(x, y) = 2^{-n}$, where $n$ is the first index with $x_n \\neq y_n$, and $d(x, x) = 0$.\nWith this metric the space is [[complete|0BY52JW6NK]], and it is [[separable|0Z2P41SVDP]], since the eventually zero sequences are dense.\nIn descriptive set theory it is written $\\mathcal{N}$ or $\\omega^\\omega$.\n\n## Remarks\n\nThe Baire space is homeomorphic to the set of [[irrational numbers|0DZXAXEJQ9]] with the topology of the real line; one homeomorphism uses continued fraction expansions.\nIt is not locally compact: no basic open set is compact, since the next term of the sequence can take infinitely many values.\nIt has a base of clopen sets.\nEvery nonempty Polish space, that is a separable completely metrizable space, is a continuous image of $\\N^{\\N}$.\nAs a complete metric space it is a [[Baire space|0ZH5AJ2ED7]] in the sense of Baire category; the two uses of the name are related but different.\n\n## Explanations\n\n### L1 human\n\nA point of the Baire space is an endless list of natural numbers.\nTwo points are close when their lists agree for a long time.\nIt is the basic space of descriptive set theory, where sets of reals are studied through the sequences that code them.\n",
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