The Simplex, the Control Set and Their Product are Compact Separable Metric Spaces

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$, and let $T>0$ be a real number. Let $\Delta^{l}$ be the probability simplex, write $|\cdot|$ for the Euclidean norm, let $d_{\mathbb{R}^{l}}$ be the Euclidean distance, a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$, and let $d_{\Delta}$ be its restriction to $\Delta^{l}$, a metric by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology.

Let $\mathcal{A}$ be a nonempty convex subset of Euclidean space $\mathbb{R}^{m}$ that is compact for the topology determined by the Euclidean distance, let $\mathcal{U}_{\mathcal{A}}$ be the set of $\mathcal{A}$-valued controls for the horizon $T$, and let $\rho$ be the metric on $\mathcal{U}_{\mathcal{A}}$ given by claim 1 of the weak metrizability and compactness theorem. Let $X=\Delta^{l}\times\mathcal{U}_{\mathcal{A}}$ be the Cartesian product and let $d_{X}$ be the product metric, a metric by claim 1 of The Product Metric is a Metric.

Then the following hold.

1. (The simplex is closed and bounded.) $\Delta^{l}$ is nonempty, $|x|\le1$ for every $x\in\Delta^{l}$, the set $\Delta^{l}$ is bounded as a subset of the metric space $(\mathbb{R}^{l},d_{\mathbb{R}^{l}})$, and $\Delta^{l}$ is closed in $\mathbb{R}^{l}$ for the topology determined by the Euclidean distance.

2. (The simplex is a compact separable metric space.) $\Delta^{l}$ is a compact subset of $\mathbb{R}^{l}$; the metric space $(\Delta^{l},d_{\Delta})$ is compact and sequentially compact; and it is separable.

3. (The control set is a compact separable metric space.) The metric space $(\mathcal{U}_{\mathcal{A}},\rho)$ is compact and sequentially compact, and it is separable.

4. (The product.) The metric space $(X,d_{X})$ is compact and sequentially compact, and it is separable. Moreover, if $D_{\Delta}$ is a countable subset of $\Delta^{l}$ that is dense in $\Delta^{l}$ for the topology of subsets open in $(\Delta^{l},d_{\Delta})$, and $D_{\mathcal{U}}$ is a countable subset of $\mathcal{U}_{\mathcal{A}}$ dense in $\mathcal{U}_{\mathcal{A}}$ for the topology of subsets open in $(\mathcal{U}_{\mathcal{A}},\rho)$, then $D_{\Delta}\times D_{\mathcal{U}}$ is countable and dense in $X$ for the topology of subsets open in $(X,d_{X})$.