Page "Embedding" Paragraph 4

In general topology, an embedding is a one-to-one function ( i. e., an injection ) that is a homeomorphism onto its image.

More explicitly, an injective continuous map f: X → Y between topological spaces X and Y is a topological embedding if f yields a homeomorphism between X and f ( X ) ( where f ( X ) carries the subspace topology inherited from Y ).

Intuitively then, the embedding f: X → Y lets us treat X as a subspace of Y.

Every embedding is injective and continuous.

Every map that is injective, continuous and either open or closed is an embedding ; however there are also embeddings which are neither open nor closed.

The latter happens if the image f ( X ) is neither an open set nor a closed set in Y.