Definition:Embedding (Topology)

Definition
Let $A, B$ be topological spaces.

Let $f: A \to B$ be a mapping.

Let the restriction $f {\restriction_{A \times f \left({A}\right)}}$ of $f$ to its image be a homeomorphism.

That is, let $f: A \to B$ be a continuous, open injection.

Then $f$ is an embedding (of $A$ into $B$).

Also known as
Some authors use the terms topological embedding, imbedding, or topological imbedding.

Some refer to an embedding in rather than an embedding into.

Also defined as
Some authors define an embedding as a continuous injection.

However, it is generally accepted that such a mapping is required to be open as well, in order for this definition to hold.

Also see

 * Continuous Injection from Compact Space to Hausdorff Space is Embedding