Almost periodic sequences and functions with given values

summary:We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set $X$ in a metric space, it is proved the existence of an almost periodic sequence $\lbrace \psi _k\rbrace _{k \in \mathbb{Z}}$ such that $\lbrace \psi _k; \, k \in \mathbb{Z}\rbrace = X$ and $\psi _k = \psi _{k + l q(k)}$, $l \in \mathbb{Z}$ for all $k$ and some $q(k) \in \mathbb{N}$ which depends on $k$