Embedding

Definition (Embedding)

An embedding of $(X,{\tau }_X)$ in $(Y,{\tau }_Y)$ is a map from $f:X\to Y$ which carries $X$ homeomorphically to its image.

For example, imagine mapping our $S^1$ circle to a trefoil knot in ${\mathbb{R}}^3$. This however, is not a homeomorphism.

Equivalently, $f$ is an embedding if:

• $f$ is injective,
• $f$ is continuous, and
• the induced map

$$f:X\to f(X)$$

is a homeomorphism, where $f(X)$ is given the Subspace Topology from $Y$.

So an embedding identifies $X$ with a subspace of $Y$ without changing its intrinsic topology.