If $\mathcal{C}$ is an operad and if $X\in\mathcal{J}$ then $CX\in\mathcal{J}$, where $\mathcal{J}$ is the category of compactly generated weakly Hausdorff spaces well-based.
I'm studying the construction of the monad asociated to a operad in "The geometry of iterated loop spaces" of J.P. May. The construction of $CX$ is as follows:
$CX$ as a set is the quotient of $\coprod_{j\geq 0}\mathcal{C}(j)\times X^j$ by the equivalence relation generated by $(c\cdot\sigma,y)\sim(c,\sigma\cdot y)$ and $(\sigma_i(c),y)\sim(c,s_i(y))$. The topology asociated with $CX$ is the union topology of $F_kCX$ where $$F_kCX:=\coprod_{j\geq 0}^{k}\mathcal{C}(j)\times X^j /\sim$$ with the quotient topology by the restricted equivalence relation.
What I know is that $\coprod_{j\geq 0}^k\mathcal{C}(j)\times X^j$ is compactly generared, so each $F_kCX$ is also a k-space, but I need to guaranty somehow that each $F_kCX$ is weak Hausdorff to modify the proof of Steenrod in "A convenient category of spaces" that if $X=\bigcup_{n\geq 0}X_n$ is the union and each $(X_{n+1},X_n)$ is a NDR-pair, and each $X_n$ is Hausdorff then $X$ is Haussdorff and $(X,x_n)$ is a NDR-pair.
I'm really confused because I know that the category used is "k-spaces weak Hausdorff" for this construction.
These notes: https://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf on CGWH don't prove what you want, but they are helpful for getting the basics down. I don't think there is any standard canonical source. When GILS was written, May used CGH, which is not so well-behaved with respect to colimits. The basic point is this: given $X$ in CGWH to prove that a quotient $p:X\to Y$ is CGWH it suffices to show that $p\times p^{-1}(\Delta_Y)$ is closed in $X\times X$. This can be applied to prove that, for example, if $G$ is a finite group acting on a CGWH space $X$, then $X/G$ is CGWH. In this case, $p\times p^{-1}(\Delta_{X/G})$ is a finite union of homeomorphic images of $\Delta_X$. Your case is proved by induction on $k$, and breaking up the space $F_k CX = F_{k-1}CX \cup (F_kCX \setminus F_{k-1}CX)$, this breaks up the diagonal, and then you get your result by induction and the case $G= \Sigma_k$ of the above result.