# Definition:Embedding (Topology)

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## Definition

Let $A, B$ be topological spaces.

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

Let the image of $f$ be given the subspace topology.

Let the restriction $f {\restriction_{A \times f\sqbrk A }}$ of $f$ to its image be a homeomorphism.

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

## Also known as

An **embedding** is also known as a **homeomorphism on its image**.

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.

## Also see

## Sources

- 1955: John L. Kelley:
*General Topology*: Chapter $4$ - 2000: James R. Munkres:
*Topology*(2nd ed.): $\S 18$