locally $\mathcal{U}$-small Dedekind real numbers for a universe$\mathcal{U}$ is defined as the Archimedean ordered integral domain$\mathbb{R}_\mathcal{U}$ with a strictly monotonic function $i:\mathbb{I}_\mathcal{U} \to \mathbb{R}_\mathcal{U}$ from the locally $\mathcal{U}$-small Dedekind real unit interval$\mathbb{I}_\mathcal{U}$ to $\mathbb{R}_\mathcal{U}$ such that $i(0) = 0$ and $i(1) = 1$.