An explicit mayer vietoris connecting form on the circle

Given: Assume countable choice. Let $U,V$ be complements of opposite circle points, let $U\cap V=W_0\bigsqcup W_1$, and let $\omega$ equal $0$ on $W_0$ and $1$ on $W_1$.

Explicit de rham mayer vietoris connecting class: Under countable choice, for a closed $k$-form $\omega$ on $U\cap V$, $\delta [\omega ]=[\eta ]$, where $\eta {\mathrm{\mid }}_U=d(-{\rho }_V\omega )$ and $\eta {\mathrm{\mid }}_V=d({\rho }_U\omega )$, with products smoothly extended by zero as in the lift construction. This class is independent of partition, lift and representative.

Zero th de rham cohomology is locally constant functions: $H_{\mathrm{d}\mathrm{R}}^0(M)$ is the algebra of locally constant real functions. For nonempty connected $M$ it is canonically $\mathbb{R}$.

Mayer vietoris sequence in de rham cohomology: Under countable choice the de Rham Mayer–Vietoris sequence is exact: $\cdots \to H^k(M)\stackrel{r^{\ast }}{\to }{H}^k(U)\oplus H^k(V)\stackrel{s^{\ast }}{\to }{H}^k(U\cap V)\stackrel{\delta }{\to }{H}^{k+1}(M)\to \cdots$, beginning with $0\to H^0(M)$.