Projective bundle associated to sum of trivial line bundles

Question

I am having some difficulty seeing that $P(1\oplus 1)=X\times S^{2}$. Here $1$ denotes the trivial (complex) line bundle over $X$, and $P$ indicates the projective bundle.

Answer 1

The total space of a trivial complex line bundle over $X$ is effectively $X \times \mathbf{C}$; "effectively" because a trivial line bundle has no "natural" trivialization, while $X \times \mathbf{C}$ has the distinguished section whose value is $1$ at each point.

In the same sense, the total space of the direct sum of two trivial complex line bundles is $X \times (\mathbf{C} \oplus \mathbf{C}) \simeq X \times \mathbf{C}^2$. Projectivizing obviously gives a global product $X \times \mathbf{CP}^1$ because the overlap maps of the rank-two vector bundle can be taken to be constant (with value the identity matrix).