Baire Space

Abstract

Next to the natural numbers, perhaps the most fundamental object of study of set theory is Baire space,

$$N{ = _{df}}(N \to N)$$

((10.1))

the set of all number theoretic sequences. If we let

$$C{ = _{df}}(N \to \{ 0,\,\,1\} )$$

((10.2))

be the Cantor set ¹ of all infinite, binary sequences, then 𝓒 ⊆ 𝓝 ⊆ (N × N) and from now familiar computations,

$$c{ = _c}{2^{{N_0}}}{ = _c}\left| {P(N)} \right|{ = _c}\left| C \right|{ \leqslant _c}\left| N \right|{ \leqslant _c}\left| {P(N \times N)} \right|{ = _c}\left| {P(N)} \right| = c$$

.