Topological embedding

Let $X$ and $Y$ be topological spaces and $f:X\to Y$ be a continuous map. $f$ is called an embedding if it is a homeomorphism on $\text{im}(f)=Z$. Slightly more rigourously, if

$$\begin{array}{rl}\hat{f}:X& ⟶Z\\ x& ⟼f(x)\end{array}$$

is a homeomorphism (restricting the domain) then $f$ is an embedding.

Examples

Consider $f:[0,1)\to \mathbb{C}={\mathbb{R}}^2$ where $f(x)=e^{2\pi x}$. This is an embedding since $f:[0,1)\to S^1$ is a homeomorphism.