Concerning the continuum hypothesis and rectilinear sections of spatial sets (Q771188)

Starting from some results of \textit{W. Sierpiński} [Fundam. Math. 38, 1--13 (1951; Zbl 0044.27301), Theorem 6] the author proves: Theorem 1: Let \(n\) be a natural number and \(S= E_1\cup E_2 \cup E_3\), a decomposition of the Euclidean space \(S\) such that every line \(\parallel x\) intersects \(E_1\) in at most \(n\) points and every line \(\parallel y\) intersects \(E_2\) in a nowhere dense set of the line; there are subsets \(P, P', P''\) of the \(xy\)-plane, each of which is everywhere of cardinality \(c\), everywhere of category II, and everywhere of positive exterior measure such that every line \(\parallel z\) intersecting \(P\) (resp. \(P'\) resp. \(P''\)) intersects \(E_2\) in a set of cardinality \(c\) (category II, positive exterior measure). \(\mathfrak A_k: S= E_1\cup E_2 \cup E_3\) and every \(x\)-line intersects \(E_1\) in a finite set, every \(y\)-line intersects \(E_2\) in \(\leq\aleph_0\) points, every \(z\)-line intersects \(E_3\) in a set of category I. The conjunction of \(\mathfrak A_k\) and \(\mathfrak K^*\) is equivalent to \(H\) (Theorem 2). Replacing in \(\mathfrak A_k\) the condition ``category I'' by ``measure 0'', resp. ``power \(<c\)'' one gets the statements \(\mathfrak A_M\), \(\mathfrak A_P\). Then \(H\Longleftrightarrow \mathfrak A_M\wedge \mathfrak M^\ast\) (Theorem 3). \(H\Longleftrightarrow \mathfrak A_P\wedge (c \)is regular) (Theorem 4) (cf. the author [Z. Math. Logik Grundlagen Math. 5, 97--116 (1959; Zbl 0086.04403)]).