Remark

The space $\mathbf{R}^0$ consists of a single point.

From Set Theory, Chapter III, § 6, no. 3, Theorem 2, Corollary I, we know that, if E is an infinite set, $E^n$ is equipotent with E for all integers $n > 0$; hence, if $n > 0$, $\mathbf{R}^n$ is equipotent with $\mathbf{R}$, that is, $\mathbf{R}^n$ has the power of the continuum (cf. Exercises 1 and 2).