Dowker-type example and Arhangelskiiʼs -property

Answering a question of M. Sakai and partially answering a question of M. Scheepers, we use the Continuum Hypothesis (CH) to show that, for a slight modification of a space constructed by Dowker in 1955, clopen point-cofinite covers can be diagonalized but there is a sequence of open point-cofinite covers that cannot be diagonalized. his space distinguishes between Arhangelskiiʼs α 2 -property for C p ( X ) and α 2 -property for C p ( X , 2 ) .