What is an inclusion map? (manifolds)

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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lavinia
Gold Member
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

Fredrik
Staff Emeritus
Gold Member
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.

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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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?

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 $\dim M = \dim N$.

lavinia