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## Main Question or Discussion Point

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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- #1

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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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lavinia

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yesIn 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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Fredrik

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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.

$$\iota \colon N \to M \colon p \to p$$

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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Why would it be open?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

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$$\iota \colon N \to M$$

is only open if ##\dim M = \dim N##.

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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.

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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