Let X be a space in which one-point sets are closed. Suppose that $\{f_α\}_{α∈J}$ is an indexed family of continuous functions $f_α : X → \mathbb R$ satisfying the requirement that for each point $x_0$ of $X$ and each neighborhood $U$ of $x_0$, there is an index α such that $f_α$ is positive at $x_0$ and vanishes outside U. Then
the function $F : X → \mathbb R^J$ defined by $F(x) = (f_α(x))_{α∈J}$ is an imbedding of X in RJ. If fα maps X into [0, 1] for each α, then F imbeds X in
$[0, 1]^J$ .