A one-dimensional Peano continuum that is not embeddable into $\mathbb{R}^3$

Question

I am looking for a continuous function $f\colon I\to X$ from the unit interval $I=[0,1]$ into a Hausdorff space $X$ such that its image $f(I)$ has dimension one and cannot be embedded into $\mathbb{R}^3$. Hausdorff spaces which are continuous images of $I$ are sometimes referred to as Peano continua. Every such space can be embedded into $\mathbb{R}^\omega$. It is not difficult to find a one-dimensional Peano continuum that can be embedded into $\mathbb{R}^2$ but not into $\mathbb{R}$ (a circle), or one that can be embedded into $\mathbb{R}^3$ but not into $\mathbb{R}^2$ (any non-planar finite graph). As any finite graph can be embedded into $\mathbb{R}^3$ (using so called book embedding), I am curious whether there exist a one-dimensional Peano continuum that cannot be embedded into $\mathbb{R}^3$.

Answer 1

Any separable metrisable space of dimension 1 (all standard dimension functions will do, covering or inductive, they all coincide for separable metric spaces, like Peano continua) has an embedding into $\mathbb{R}^3$. (This is Nöbeling's embedding theorem). It can be generalised to $n$ and $2n+1$ in fact. See Engelking, Theory of Dimensions, finite and infinite Thms 1.11.4 and 1.11.5.