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Some addition theorems in a class of bicompacta. (English. Russian original) Zbl 0541.54004
Sib. Math. J. 24, 934-941 (1983); translation from Sib. Mat. Zh. 24, No. 6(142), 135-143 (1983).
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).
##### MSC:
 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54D30 Compactness 54B05 Subspaces in general topology
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##### References:
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