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Fellowroot
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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?
yesFellowroot 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?
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
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.
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.
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.
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.
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.