On partition of topological spaces. (English) Zbl 0539.54005

Bregman, Yu. K.; Shapirovskij, B. Eh.; Shostak, A. P.

Čas. Pěst. Mat. 109, 27-53 (1984).

The authors study the problem of the existence of a Hausdorff space Y such that, for each \(A\subset Y\), the Cantor discontinuum is (homeomorphic to) a subspace of A or of \(Y-A.\) They make significant contributions to this problem. Theorem. [ACP#]\(\vee [V=L\)]. For every topological space X there exists a partition \({\mathcal R}=\{R_{\alpha}:\quad \alpha<c\}\) such that for every closed regular countably compact subset F without isolated points the intersections \(R_{\alpha}\cap F\) are everywhere dense in F for all \(R_{\alpha}\) (thus \({\mathcal R}=\{R_{\alpha}:\quad \alpha<c\}\) is a decomposition for all such F simultaneously). In particular, for every Hausdorff compact space X without isolated points there exists a (universal) decomposition \({\mathcal R}=\{R_{\alpha}:\quad \alpha<c\}\) such that if F is a closed subset without isolated points then \([F\cap R_{\alpha}]=F\) for every \(\alpha<c. (ACP\#\equiv\) ”For every cardinal \(\mu \geq c\) there exists cardinal \(t<c\) such that \(\mu^{\aleph}\leq \mu_ t,\) and \(\aleph_ 1<c.'')\) Theorem. In every topological space there are subspace \(X_ 0\) and \(X_ 1\) such that \(X=X_ 0\cup X_ 1\) and neither \(X_ 0\) nor \(X_ 1\) contain a closed countably compact regular unscattered subspace F (in X). In particular, if X is Hausdorff, then every compactum contained in \(X_ 0\) or \(X_ 1\) is scattered.

Reviewer: C.R.Borges

MSC:

54C25	Embedding
54A35	Consistency and independence results in general topology

Keywords:

scattered spaces; Cantor discontinuum; ACP#; \(V=L\); partition; closed regular countably compact subset; isolated points; (universal) decomposition; closed countably compact regular unscattered subspace