An injective continuous map f : X → Y between topological spaces such that f is a homeomorphism onto its image f(X) endowed with the subspace topology from Y; equivalently X is topologically identified with the subspace f(X) of Y.

Principle

Preserve both the point-set injectivity and the intrinsic topology: the map must not only be one-to-one and continuous but must reflect open (or closed) sets so that topological properties of X are exactly those inherited from its image in Y.

Demonstration

The inclusion of the circle S^1 as the unit circle in R^2 is a topological embedding: the inclusion map is injective, continuous and its inverse from the image (with the subspace topology) is continuous, so S^1 is homeomorphic to its image.

Misapplication

Calling every injective continuous map an embedding while neglecting that an injective continuous map may fail to be a homeomorphism onto its image (for example when the image has a strictly coarser subspace topology than the domain), or conflating embedding with immersion in the smooth category.

Consequence

When a map is an embedding, one may treat the domain as a subspace of the codomain and transfer local and global topological invariants, use subspace constructions and apply embedding-dependent theorems (e.g., tubular neighborhood results in manifold contexts when additional structure exists).

Reversal

The opposite notion is a continuous surjection or quotient map that identifies points rather than embedding: such maps often destroy separation properties and cannot be used to regard the domain as a topological subspace of the codomain.

Boundary

Depends on the topology of the codomain: embeddings may have non-closed images, and in some contexts one requires additional conditions (closed embedding, proper embedding, smooth embedding) which are stronger; embeddings are a property of the map, not solely of the pair of spaces.

Semantic Tension

Tension exists between embedding, homeomorphism and immersion: a homeomorphism is a bijective embedding whose inverse is continuous on the whole codomain, while an immersion (in differential topology) concerns derivatives and need not be injective or topology-preserving; precise context matters.

Synthesis

A topological embedding is an injective continuous map that identifies the domain with a subspace of the codomain by being a homeomorphism onto its image, enabling one to regard the source as literally sitting inside the target while preserving its topology.