We investigate the minimal width of deep neural networks that can approximate all continuous functions from Rⁿ to Rᵐ in the topology of uniform convergence on compact subsets. We show that whenever the real projective space RPⁿ⁻¹ admits a smooth immersion into Rⁿ, the minimal width is at least n+min{n,m}, assuming that the activation function is continuous and monotone. As a consequence, the minimal width of ReLU networks is n+m whenever n is equal to 2, 3, 4, 7 or 8 and 1≤ m ≤ n, extending results of Hanin-Sellke and Hwang. This suggests an intriguing connection between the approximation theory of neural networks and differential topology.