What is an inclusion map? (manifolds)

In summary: The inclusion map is a map that takes a set and maps it to a subset. It is typically represented by a curved arrow and 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 is open, injective, and an immersion. This means that the inclusion map is open, meaning that it preserves the topology of the subset, and injective, meaning that it is a one-to-one map. This also means that the differential of the inclusion map has a trivial kernel at every point. In general, the inclusion map is used to define embedded submanifolds, where a subset is given by
  • #1
Fellowroot
92
0
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
Fellowroot said:
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
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
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
Geometry_dude said:
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
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
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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What is an inclusion map?

An inclusion map is a type of function in mathematics that maps one set into another set. It is often used in the context of manifolds, which are mathematical objects that locally resemble Euclidean space.

How does an inclusion map work?

An inclusion map works by taking elements from one set and mapping them into another set. It preserves the structure of the original set and is usually defined as the identity on the subset that is being mapped.

What is the purpose of an inclusion map?

The purpose of an inclusion map is to embed a smaller set into a larger set. This allows for the study of the smaller set in the context of the larger set, and can provide insights and connections between different mathematical objects.

Are all inclusion maps the same?

No, not all inclusion maps are the same. They can vary depending on the specific sets and context in which they are used. However, they all follow the same general concept of mapping one set into another.

Can inclusion maps be used in any type of mathematical object?

Inclusion maps are commonly used in manifolds, but they can also be used in other mathematical objects such as topological spaces and groups. As long as there is a clear structure and relationship between two sets, an inclusion map can be defined and used.

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