Is every GO-space collectionwise normal? Is every GO-space collectionwise normal? And what is the relation between collectionwise normal and monotonically normal? I know a GO-space is always monotonically normal. Thanks.
 A: For your second question, monotonically normal spaces are collectionwise normal.
  (And this answers your first question.)

Suppose that $\mathcal{H}$ is a discrete family of closed subsets of $X$.  Note that for $H \in \mathcal{H}$ the set $\hat{H} = \bigcup \{ K \in \mathcal{H} : K \neq H \}$ is a closed set disjoint from $H$, and so consider $U_H = D_X ( H , \hat{H} )$.  (Where $D_X$ denotes the monotone normality operator on $X$.)
Given distinct $H , H^\prime \in \mathcal{H}$ by the monotonicity of $D_X$ it follows that $U_H \cap U_{H^\prime} = D_X ( H , \hat{H} ) \cap D_X ( H^\prime , \hat{H}^\prime ) \subseteq D ( H , H^\prime ) \cap D ( H^\prime , H ) = \emptyset$, and so the family $\{ U_H : H \in \mathcal{H} \}$ is pairwise disjoint.


Information about monotonically normal spaces can be found in  

R. W. Heath, D. J. Lutzer and P. L. Zenor, Monotonically normal spaces,  Trans. Amer. Math. Soc. 178 (1973), 481-493, MR0372826; link

The above is essentially a re-working of Theorem 3.1 of this paper.
