This is a follow-up to Dan Ramras' answer of this question.

The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).

The weak Hausdorff rather than the Hausdorff property should be required of spaces [...] in order to validate some of the limit arguments used [...].

I was not able to figure out which type of "limit arguments" really would have needed the weak Hausdorff condition instead of the regular one used in the original paper and would be happy to understand the necessity of the correction, ideally by means of an explicit example in the paper.

CG spaces) is stated entirely in terms of the CG category, whereas the H condition (the diagonal is closed when the product is taken inTop) refers us back to Top. So there's a "mismatch" in the definition of CGH -- it's like defining a scheme to be separated if its underlying space is Hausdorff, which is totally wrong. Plus, we're potentially re-importing pathologies from Top... $\endgroup$