Topological embedding

Definition

Let $X$ and $Y$ be topological spaces. A function $f \colon X \to Y$ is a topological embedding when $f$ is injective and the map $X \to f(X)$, $x \mapsto f(x)$, is a homeomorphism. Here the image $f(X)$ has the subspace topology of $Y$.

Remarks

Each embedding is continuous, because the inclusion of a subspace is continuous. The inclusion of a subspace is an embedding. The map $\mathbb{R} \to \mathbb{R}^2$, $x \mapsto (x, 0)$, is an embedding. The map $t \mapsto (\cos t, \sin t)$ from $[0, 2\pi)$ to $\mathbb{R}^2$ is continuous and injective, but not an embedding. A continuous injective map from a compact space to a Hausdorff space is an embedding.

Explanations

L1 human

An embedding puts a copy of $X$ inside $Y$ without changing its shape. An injective continuous map can still bring the ends of the space together in the limit. Then it is not an embedding.