Baire space

Any space in which the Baire theorem on complete spaces is valid.

The metric space the points of which are infinite sequences $\{n_i\}=\{n_1,n_2,\dotsc\}$ of natural numbers, and the distance is given by the formula:

$$\rho(\{n_i\},\{m_i\}) = \frac1{k_0}$$

where $k_0$ is the first natural number $k$ for which $n_k\neq m_k$. This is a complete metric separable zero-dimensional space containing the topological image of any metric separable zero-dimensional space.