Does embedding a rectangle in a plane have additional properties compared to embedding a segment?

Question

Let $f: [0, 1] \times [-ε, + ε] \to \mathbb{R}^2$ a topological embedding. Does the restriction of $f$ to $[0, 1] \times \{0\}$ have any additional properties in comparison with the topological embeddings of the segment $f: [0, 1] \to \mathbb{R}^2$?