The compact spaces are not enough by themselves to give a measure to the given spaces, the generalization of these, the paracompact spaces allow to provide the desired metric to work on the space, it is therefore necessary to know how to share and the most important properties of the paracompact space, for example, that the product of the paracompact spaces is not paracompact, but that by adding the hypothesis that at least one of the spaces is compact, the product will finally be paracompact.
