Families of rank one in topological spaces
by
Mitrofan M. Choban
Tiraspol State University, Kishinev, Republic of Moldova

All considered spaces are assumed to be regular. A family B of subsets has the rank one if for every U, V in B we have U \cap V = \emptyset, or U subset or equal V, or V subset or equal U. The rank r(X) of a space X is the smallest cardinal m such that there exists a T₀-separating family B of open-and-closed subsets of X which is a union of m subfamilies of rank one. For a space X by \psi(X), c(X), d(X), l(X), hl(X), ml(X) and |X| denote the pseudocharacter, the Souslin number, the density, the Lindelöf number, the hereditarily Lindelöf number, the metalindelöf number and the cardinality of X respectively. We consider that m + \infty = \infty+ m = \infty and m < \infty for every cardinal m.

Theorem 1. \psi(X) <= c(X) + r(X).

Theorem 2. |X| = hl(X) = \psi(X) + l(X) <= d(X) + l(X) + r(X) = d(X) +ml(X) + r(X) and d(X) = c(X) for every scattered space X.

Corollary 1. |X| = d(X) = c(X) for every metalindelöf scattered space X of countable rank.

Theorem 3. Every hereditarily paracompact scattered space has the rank one.

Theorem 4. Every space X is an open continuous image of some perfectly normal paracompact \sigma-discrete space Y of rank one. If X is scattered then Y is scattered too.

Theorem 5. If Y is a closed continuous image of a scattered space X then |Y| <= d(Y) + r(X) + ml(X).

Theorem 6. Let X be the Alexandroff compactification of an uncountable discrete space. Then r(Xⁿ) = n for every natural number n.

Theorem 7. Let m = |N| and m <= n < \lambda <= n^m. Then there exists a compact scattered space X(n, \lambda) for which d(X(n, \lambda)) = n and r(X(n, \lambda)) = \lambda.

Corollary 2. There exists a separable scattered compact space Z of uncountable rank. Moreover, if Z is a continuous image of a compact scattered space X then the rank of X is uncountable.

Theorem 8. Let X be a suborderable space. Then r(X) = 1 iff r(X) < \infty.

The results give answers to some questions posed by S. Troyanski, P. Kenderov and S. Nedev.

Date received: June 19, 2000