  The study of universal approximation properties (UAP) for neural networks
(NN) has a long history. When the network width is unlimited, only a single
hidden layer is sufficient for UAP. In contrast, when the depth is unlimited,
the width for UAP needs to be not less than the critical width
$w^*_{\min}=\max(d_x,d_y)$, where $d_x$ and $d_y$ are the dimensions of the
input and output, respectively. Recently, \cite{cai2022achieve} shows that a
leaky-ReLU NN with this critical width can achieve UAP for $L^p$ functions on a
compact domain ${K}$, \emph{i.e.,} the UAP for $L^p({K},\mathbb{R}^{d_y})$.
This paper examines a uniform UAP for the function class
$C({K},\mathbb{R}^{d_y})$ and gives the exact minimum width of the leaky-ReLU
NN as $w_{\min}=\max(d_x,d_y)+\Delta (d_x, d_y)$, where $\Delta (d_x, d_y)$ is
the additional dimensions for approximating continuous functions with
diffeomorphisms via embedding. To obtain this result, we propose a novel
lift-flow-discretization approach that shows that the uniform UAP has a deep
connection with topological theory.
