The fact that the topological dimension of $\mathbb{R}^n$ is $n$ implies that there is no continuous injection of $\mathbb{R}^n$ into $\mathbb{R}^{n-1}$. Does this statement survive if the requirement of injectivity is relaxed to being injective on a large set? Alex Izzo showed that the answer is negative when the largeness is measure-theoretic, more precisely, when the function is required to be injective only on a conull set. What about the notion of largeness provided by Baire category? Namely, does there exist a continuous function from $\mathbb{R}^n$ to $\mathbb{R}^{n-1}$ that is injective on a comeager set? We will discuss this and related questions, as well as some partial results.