Suppose I have a topological space $X$ with a collection of closed subsets $X_\tau$ for $\tau \in P$ where I think of $P$ as a poset with $\tau \leq \lambda \iff X_\tau \subset X_\lambda$. Is there some nice (algorithmic?) way to get information about the (compactly supported) cohomology of $U_{\tau} = X_\tau - \bigcup_{\lambda<\tau}X_\lambda$ in terms of the cohomology of the $X_\tau$?
For instance, if we I just one closed subset $Z \subset X$, then there is a long exact sequence: $$\dots \to H^n_c(U) \to H^n_c(X) \to H^n_c(Z) \to \dots$$ Is there something similar in the general case? I can certainly think of some inductive process for building up the relevant cohomology starting from the lowest strata by a combination of the excision sequence and Mayer-Vietoris but it would be nice if there was succinct description.