How can we show that given the manifold $M=\mathbb{S}^2$ and $U=\mathbb{S}^2\setminus \{A\}$ and $V=\mathbb{S}^2 \setminus \{B\}$, then:
$$H_{dR}^1(U)\oplus H_{dR}^1(V)\rightarrow H_{dR}^1(U\cap V) $$ cannot be surjective. The reasoning in my book says that $ H_{dR}^1(U)=H_{dR}^1(V)=\{[0]\}$ and $H_{dR}^1(\mathbb{S}^1)=H_{dR}^1(U\cap V)=\mathbb{R}$, hence they cannot be surjective. But I am having trouble seeing why this is the case, as in why the above is equal to $\mathbb{R}$ and $\{[0]\}$.
Can anyone help me understand my confusion? Thank You