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We introduce the notions of a path complex and its homologies. Particular cases of path homologies are simplicial homologies and digraph homologies. We state and prove some properties of path homologies, in particular, the K¨unneth formulas for Cartesian product and join, which happen to be true at the level of chain complexes.

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We find some necessary conditions for a real Banach space to be an almost CL-space. We also discuss the stability of CL-spaces and almost CL-spaces by$c$_{0}- and$l$_{1}-sums. Finally, we address the question if a space of vector-valued continuous functions can be a CL-space or an almost CL-space.