Quarter-stratifiable GO-spaces
by
David Lutzer
College of William and Mary, Williamsburg, VA 23187-8795
Coauthors: Harold Bennett, Texas Tech University, Lubbock, TX 79409

To extend classical results of W. Rudin, Kuratowski, and Montgomery, T.O. Banakh defined that a space X is quarter-stratifiable if for each n there is an open cover { g(n, x) : x in X } such that whenever y is in g(n, x(n)) for each n, the sequence x(n) converges to y. Note that g(n, x) is not required to contain x. For example, letting g(n, x) = (x - 1/n, x - 1/(2n)) we obtain a quater-stratification for the Sorgenfrey line.

In this paper (to appear in Proc. Amer. Math. Soc.), we study quarter-stratifiable generalized ordered (GO) spaces. We show that any quarter-stratifiable GO-space X has a G_d-diagonal and has a s-closed-discrete dense subset (hence d(X) = c(X) and X is perfect). We show that a perfect GO-space X with a G_d- diagonal is quarter-stratifiable if and only if there is a s- closed-discrete subset Q of X such that in the subspace X - Q there are disjoint relatively F_s-subsets A and B with the property that A contains R(X) - Q and B contains L(X) - Q, where R(X) is the set of non-isolated points x in X for which the half-line { y in X: x is less than or equal to  y } is open in X, and where L(X) is analogously defined. We deduce that

(1) any subspace of a quarter-stratifiable GO-space is quarter-stratifiable;

(2) if a GO-space X has a s-locally-finite cover by quarter- stratifiable closed subspaces, then X is quarter-stratifiable;

(3) if a perfect GO-space X can be written as the union of a metrizable subspace and a quarter-stratifiable subspace, then X is quarter-stratifiable;

(4) any GO-space constructed on a Q-set in the real line will be quarter-stratifiable.

Date received: February 7, 2005