Weakly perfect generalized ordered spaces (Q2725452)

Weakly perfect spaces were introduced by the reviewer [Mat. Vesn. 35, 145-150 (1983; Zbl 0539.54002)] in the following way: a space \(X\) is weakly perfect if every closed set \(A\subset X\) contains a set \(B\) which is dense in \(A\) and a \(G_\delta\)-set in \(X\). A consistent example of weakly perfect compact space \(X\) that is not perfect was constructed. Later, \textit{R. W. Heath} [Publ. Inst. Math., Nouv. Sér. 46(60), 193-195 (1989; Zbl 0694.54021)] constructed such ZFC examples. The authors of this paper study weak perfectness in linearly ordered and generalized ordered spaces. The main results are: (1) there are compact linearly ordered spaces which are hereditarily weakly perfect but not perfect; (2) for each subset \(P\subset [0,1]\) a space \(X(P)\) is constructed and it is shown that it is (compact and linearly ordered) weakly perfect but not perfect iff \(P\) is an uncountable perfectly meager set; (3) there are Michael line-type subspaces of [0,1] which are GO hereditarily weakly perfect but not perfect spaces. A new interesting internal characterization of perfectly meager subsets of [0,1] is also given.