On duality for cohomology with compact supports

The canonical mapping of cohomology with compact supports via linear functionals on homology, generally speaking, is not surjective. The image of the mapping is described by linear functionals with compact supports. In this paper we prove the formula $H_c ^ k (X; \mathbb {R}) = {\rm Hom}_c (H_k ^ c (X; \mathbb {R}), \mathbb {R})$, where $ X $ is a countable simplicial complex with an additional requirement, the star of each simplex has a finite number of simplices in all dimensions.