A web search suggests that the category of *CGWH* spaces was introduced in the paper [*"Classifying Spaces and Infinite Symmetric Products"*][1] by M. C. McCord (Transactions of the American Mathematical Society Vol. 146, (1969), pp. 273-298). 

 McCord motivated introduction of his "weak Hausdorff" separation axiom by noting that

> "the requirement of the Hausdorff condition can be a problem
because certain standard operations on spaces can lead outside the category", 
in particular quotient spaces in algebraic topology and topological algebra.


  [1]: https://www.jstor.org/stable/1995173