Why paracompact spaces are required to be Hausdorff

Question

If paracompactness is supposed to be a generalization of compactness. Why is Huasdorfness required in its definition?

It seems like it is more a generalization of compact normal spaces. But the name does not suggest so.

Answer 1

In fact, many texts require compact spaces to be Hausdorff too, making them normal as well. Paracompact spaces that are Hausdorff are also normal, which is a generalisation of that fact. Indeed paracompactness can be seen as a strong form of normality (e.g. normal spaces obey "every point finite open cover has a point-finite shrinking", so normality behaves like a weak "covering property" (as properties like compactness, paracompactness, Lindelöfness, etc. are called as a category).

Historically paracompactness grew out of a common generalisation of metrisability and compactness (it occurs quite naturally in proving metrisation theorems), so in that light adding Hausdorff seems natural, as we then get the normality people were already used to getting in compactness anyway. Many alternative characterisations of paracompactness require regularity as well. Most spaces that occur "in nature" that are paracompact were already Hausdorff anyway. And Hausdorffness ensures that we have lots of open sets (enough to separate points) and so lots of open covers, so that paracompactness "means" something, intuitively speaking.