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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:
  A. V. Arkhangel’skii, ?On compact spaces which are unions of certain collections of subspaces of special type,? Comment. Math. Univ. Carolinae,17, No. 4, 737-753 (1976). · Zbl 0338.54015  A. V. Arkhangel’skii, ?Topologies admitting a weak connection with orderings,? Dokl. Akad. Nauk SSSR,238, No.4, 773-776 (1978).  M. G. Tkachenko, ?The behavior of cardinal invariants when taking unions of chains of subspaces,? Vestn. Mosk. Gos. Univ., No. 4, 50-58 (1978).  M. G. Tkachenko, ?Bicompacta which can be represented in the form of unions of a countable number of left subspaces,? Comment. Math. Univ. Carolinae,20, No. 2, 361-395 (1979).  B. É. Shapirovskii, ?Mappings on Tikhonov cubes? Usp. Mat. Nauk,35, No. 3, 122-130 (1980).  R. Pol and E. Pol, ?Remarks on Cartesian products,? Fundam. Math.,93, 57-69 (1976). · Zbl 0339.54008  B. É. Shapirovskii, ?Canonical sets and character. Density and weight in bicompacta,? Dokl. Akad. Nauk SSSR,218, No. 1, 58-61 (1974).
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