Some addition theorems in a class of bicompacta (Q794308)

Let X be a compact space which is the union of at most \(\tau\) many ''good'' subspaces with infinite \(\tau\). Suppose that X is the union of some family \(\gamma\) of its subspaces such that \(| \gamma | \leq \tau\) and the tightness of M is less than \(\tau\) for each \(M\in \gamma\). If \(\tau\) is regular, then t(X)\(\leq \tau\) and there exists a point \(x\in X\) with \(\pi \chi(x,X)<\tau\) (Theorems 1.1 and 2.5). If we suppose now that each \(M\in \gamma\) is an \(\alpha\)-left subspace of X (instead of \(t(M)<\tau)\), then there exists a point \(x\in X\) with \(\pi \chi(x,X)<\tau\), too, and the regularity of \(\tau\) is not essential (Theorem 2.2). Consequently, any compact space which is representable as the union of some countable family of its \(\alpha\)-left subspaces, is scattered. This generalizes some of the author's and I. Juhász's results. Now let \(\tau\) be an infinite cardinal with \(2^{\lambda}\leq \tau\) for each \(\lambda<\tau\), and a compact space X be the union of some family \(\gamma\) of its \(\alpha\)-left subspaces with \(| \gamma | \leq \tau\). Then there exists a point \(x\in X\) such that \(\chi(x,X)<\tau\) (Theorem 2.3). It is not known whether the condition \(''2^{\lambda}\leq \tau\) for each \(\lambda<\tau ''\) can be excluded from the previous theorem. Theorem 2.2 has a more general version for the case of a regular uncountable cardinal \(\tau\). Namely, let a compact space X be the union of some family \(\gamma\) of its \(\alpha\)-expanded subspaces and \(| \gamma | \leq \tau\), where \(\tau\) is regular and uncountable. Then there exists a point \(x\in X\) such that \(\pi \chi(x,X)<\tau\) (Theorem 2.4).