Separation of diagonal in monotonically normal spaces and their products

As separation of diagonal, we study when monotone normality implies Δ-paracompactness or Δ-normality. For that, it is proved that every monotonically normal space is Δ-paracompact if the projection of its square is closed. Moreover, it is proved that every monotonically normal space is Δ-normal if it has countable tightness (or countable extent). In particular, the parenthetic part is an affirmative answer to Burke and Buzyakova's problem in 2010. Secondly, we study the relation between normality and Δ-paracompactness or Δ-normality in certain products. For that, we additionally introduce two new neighborhood properties. Using these ones, it is proved that the product X×K of a monotonically normal space X and a compact space K is Δ-paracompact (respectively, Δ-normal) if and only if X is Δ-paracompact (respectively, Δ-normal) and X×K is normal.