Long exact sequence for cohomology with compact supports

$H_c(X)=\tilde H(X^*)$ where $X^*$ is the one-point compactification of $X$ — so theorems about cohomology with compact can be deduced from theorems about ordinary cohomology.

In particular, the long exact sequence for the pair $(X^*,C^*)$ in ordinary cohomology gives the desired exact sequence ($H(X^*,C^*)\cong H_c(U)$ by excision).

Anyway, the map $C_c(X)\to C_c(C)$ is the usual restriction. And the kernel of this restriction is $C_c(X,C)$ which is quasiisomorphic to $C_c(U)$ (by excision: $C(X,(X-K)\cup C)\cong C(U,X-(K\cap U))$; cf. «extension by zero» in the de Rham case).

The existence of such an exact sequence can be found in Bredon's book "Sheaf Theory" (version McGraw-Hill 1967, section III.1) for the singular cohomology with compact support where one has to assume that the spaces $X$ and $C$ are HLC which means the following: for each $x \in X$ and neighborhood $U$ of $x$ there is a neighborhood $V \subseteq U$ of $x$ depending on $p$ such that $H_p(V) \to H_p(U)$ is trivial where $H_∗$ is the reduced singular homology.

Another proof can be found in Spanier's book "Algebraic Topology" (version McGraw-Hill 1966, section 6.7), for the Alexander cohomology with compact support, assuming the $X$ is a locally compact Hausdorff space. (In fact, one has to look at the proof of Theorem 15.)

Note that HLC implies that Alexander cohomology coincides with the singular cohomology. Also note that CW-complexes are HLC since they are locally contractible. So the sequence exists for example for complex algebraic varieties and closed subvarieties.