Some properties involving feeble compactness, IV: Compactness-like properties defined via dense subspaces

We study several compactness-like properties arising from the existence of dense subspaces satisfying relative compactness conditions, namely: countably pracompact, totally countably pracompact, densely ω-bounded, and sequentially pracompact spaces. These classes refine classical notions such as sequential compactness and ω-boundedness and admit a natural hierarchy. We establish various preservation results for these properties under perfect open mappings and product spaces. In the context of topological groups, we prove that if H is a locally compact subgroup of a topological group G, then the corresponding quotient mapping $\pi :G\to G/H$ allows the transfer of local versions of these properties from $G/H$ to G. We also analyze the extent to which these properties satisfy the three-space property and introduce the class of PC-spaces to characterize when such transfer is possible. Finally, we address structural questions on densely ω-bounded paratopological groups and provide conditions under which they are topological.