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[parent] inclusion mapping (Definition)

Definition Let $ X$ be a subset of $ Y$. Then the inclusion map from $ X$ to $ Y$ is the mapping

$\displaystyle \iota: X$ $\displaystyle \to$ $\displaystyle Y$  
$\displaystyle x$ $\displaystyle \mapsto$ $\displaystyle x.$  

In other words, the inclusion map is simply a fancy way to say that every element in $ X$ is also an element in $ Y$.

To indicate that a mapping is an inclusion mapping, one usually writes $ \hookrightarrow$ instead of $ \to$ when defining or mentioning an inclusion map. This hooked arrow symbol $ \hookrightarrow$ can be seen as combination of the symbols $ \subset$ and $ \to$. In the above definition, we have not used this convention. However, examples of this convention would be:

  • Let $ \iota:X\hookrightarrow Y$ be the inclusion map from $ X$ to $ Y$.
  • We have the inclusion $ S^n\hookrightarrow \mathbb{R}^{n+1}$.



"inclusion mapping" is owned by Koro. [ owner history (1) ]
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See Also: pullback

Other names:  inclusion map, inclusion

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Cross-references: arrow, mapping, subset
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This is version 6 of inclusion mapping, born on 2003-06-26, modified 2004-01-31.
Object id is 4402, canonical name is InclusionMapping.
Accessed 9382 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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inclusion map on Riemannian manifolds by ulrich_utiger on 2006-04-12 13:52:45
My professor is pretending that the inclusion map i:M C Rn takes the points of definition (x1,...,xm) of a parametrization x:U -> M (where M C Rn is a manifold of dimension m and U an open set of Rm) to Rn in the following sense: if m<n and p=x(q) is a point of M then i(p)=(x1,...,xm,0,...,0) is a point of Rn. This definition gives no sense to me. But I dont dare asking him again about this. Can anyone explain the sense of this definition and how it is compatible with the definition on this site?
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Why is inclusion map a topological embedding? by zhaoway on 2003-11-23 09:32:47
I read in John M. Lee's book: Introduction to topological manifolds, proposition 3.4 (a) says that inclusion map is a topological embedding. But let's suppose there is an inclusion map from a closed subset A of X to X, then in the subspace of A inherited from X, A itself is an open set, while its image after the inclusion map is a closed subset in X. Henceforthly, this inclusion map between A and its image is not a homeomorphism. Hence no topological embedding. Where have I gone wrong? Please! Thank you!
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