In Tu's An Introduction to Manifolds, he mentions: "If $U$ is an open subset of a manifold $M$, then the inclusion $i:U\to M$ is both an immersion and a submersion."
Is the statement true because $\dim TU = \dim TM$ and by dimensionality argument $di_p$ is both injective and surjective?
Is the inclusion, $i$, also an embedding?
Is the inclusion $i$ ever stated or used explicitly in calculations? For example, a chart on a manifold $M$ is $\phi : U\subset M \to \mathbb{R}^n$ so if $\Phi : M \to \mathbb{R}^n$, the inclusion is used to restrict $\Phi$ from a global chart to a local chart $\phi = \Phi \circ i$.
Assuming that $U\cap V\neq \emptyset$, and $U,V$ open sets that cover $M$, is the inclusion $j_U : U\cap V \to U$ an immersion, submersion, or embedding? Is $j_U$ ever used explicitly in calculations?
What does Hirsch mean by saying that "...a manifold is built from elementary objects (open sets) "glued" together by maps of specified kind (diffeomorphisms).."
Wlog we can suppose that $U$ is the domain of a chart: $\varphi: U\longrightarrow U'\subset\Bbb R^n$ and: $$\varphi\circ i\circ\varphi^{-1}: U'\longrightarrow U'$$ is the identity in a open subset of $\Bbb R^n$, so $i$ is local diffeomorphism $\implies$ for all $p\in U$ $di_p$ is bijective (injective and surjective) $\implies$ $i$ is immersion and submersion.
Also $i$ is an embedding = injective immersion which is homeomorphism onto its image. And by the same reasons $j_U$ is an immersion/submersion/embedding.
means that a manifold can be considered a quotient. Example: consider $S^1 = \{(x,y)\in\Bbb R^2\,\vert\,x^2 + y^2 = 1\}$. $S^1\setminus\{(0,1)\}$ and $S^1\setminus\{(0,-1)\}$ are homeomorphic to intervals. Take the disjoint union of both intervals and identify points with the same image in $S^1$.