What is an inclusion map? (manifolds)

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In my book its says let i: U →M (but with a curved arrow) and calls it an inclusion map. What exactly is an inclusion map? Doesn't the curve arrow mean its 1-1? So are inclusion maps always 1-1?
 

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  • #2
lavinia
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In my book its says let i: U →M (but with a curved arrow) and calls it an inclusion map. What exactly is an inclusion map? Doesn't the curve arrow mean its 1-1? So are inclusion maps always 1-1?
yes
 
  • #3
Fredrik
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The map ##I:X\to X## defined by ##I(x)=x## for all ##x\in X## is called the identity map on ##X##. If X is a subset of Y, then the map ##J:X\to Y## defined by ##J(x)=x## for all ##x\in X## is called the inclusion map from ##X## to ##Y##. If X=Y, the identity map and the inclusion map are the same. If you're using a definition of "map" such that the codomain isn't one of the things that identify the map, then the identity map and the inclusion map are the same, even when X is a proper subset of Y.
 
  • #4
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The curved arrow is usually reserved for inclusions. In general, if you have a differentiable manifold ##M## and a subset ##N \subseteq M## that is also a differentiable manifold then the inclusion map
$$\iota \colon N \to M \colon p \to p$$
is open (trivially, with respect to the subspace topology on the image), injective and an immersion (i.e. the differential ##\iota_*## has trivial kernel at every point). The tuple ##(N, \iota)## is an embedded submanifold of ##M## and one can always think of embedded submanifolds as being given by subsets by looking at the image and the inclusion map.
 
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  • #5
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The curved arrow is usually reserved for inclusions. In general, if you have a differentiable manifold ##M## and a subset ##N \subseteq M## that is also a differentiable manifold then the inclusion map
$$\iota \colon N \to M \colon p \to p$$
is open
Why would it be open?
 
  • #6
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It is open with respect to the subspace topology on the image, which just happens to be the topology on the set itself by definition. So actually the map
$$\iota \colon N \to M$$
is only open if ##\dim M = \dim N##.
 
  • #7
lavinia
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A subset of a set can be viewed as a set in itself. The inclusion map takes its points as a set and maps them to the corresponding points in the subset.

For instance the inclusion of {a} into {a,b} maps {a} to the subset {a} ⊂ {a,b}.

The idea of inclusion applies to sets and subsets. It is completely general and is not restricted to manifolds or topological spaces.
 
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