In the image included here (from Dow's Introduction to Applications of Elementary Submodels) absoluteness is used to argue that if M "thinks" that B is a base for $(X,\tau)$ then $H(\Theta)$ does too. But doesn't this follow from elementarity? And doesn't this also imply that $H(\Theta)$ thinks $B$ is point countable as well? Why aren't the lines in pink and green superfluous?