[[Mathematics MOC]]
# Embedding

An **embedding** is an [[Surjectivity, injectivity, and bijectivity|injective]] [[Morphism|homomorphism]] which induces an [[Morphism|isomorphism]] with its image.
See [[Regular monomorphism]] for a categorical generalization.

## Topology
An **embedding** is an injective continuous map $f : Y \hookrightarrow X$ such that it would be impossible for $Y$ to have a [[Coarseness and fineness of topologies|coarser]] topology,
i.e. the topology of $Y$ is the same as the [[Subspace topology]] induced by $f$.[^br] #m/def/topology 

[^br]: 2020, [[@bradleyTopologyCategoricalApproach2020|Topology: A categorical approach]], 26

## Differential topology

A $C^k$ **embedding** of $Y$ in $X$ is a $C^k$ [[diffeomorphism]] between $Y$ and a [[submanifold]] of $X$. #m/def/geo/diff

## Others

- [[Category embedding]]

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