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Organizers |
GO-spaces with S\delta-diagonals and dense S\delta-diagonals
by
Masami Hosobuchi
Tokyo Kasei Gakuin University
| Abstract |
A concept of S\delta-diagonal was introduced by R. H. Bennett to study the quasi-developability of linearly ordered topological spaces.
Definition. A subset A of a space X is called an S\delta-subset if there exists a collection {U(n):n in N} of open subsets of X such that, for p in A and q in X \A, there exists an m in N such that p in U(m) and q not in U(m). A space X has an S\delta-diagonal if the diagonal subset \Delta of X ×X is an S\delta-subset.
Theorem 1. Let X be a topological space. Then X has an S\delta-diagonal if and only if there exists a family {G(n):n in N} of countable collections of open subsets of X such that, for three points x, y and z, where y =/= z, there exists an m in N such that x in \cup G(m) and no elements of G(m) contain {y, z}.
Corollary. If X has a G\delta-diagonal, then X has an S\delta-diagonal. If X has an S\delta-diagonal, then X has a quasi-G\delta-diagonal.
Theorem 2. Let X be a GO-space with an S\delta-diagonal. If R \cup L is countable, then X* has an S\delta-diagonal. If, furthermore, for x in R (respectively, y in L), there exists an increasing sequence {xn} (resp. a decreasing sequence {yn}) such that sup{xn}=x (resp. inf{yn}=y), then L(X) has an S\delta-diagonal.
Definition. A Hausdorff space X has a dense S\delta-diagonal if there exists a dense subset of the diagonal \Delta that is an S\delta-subset of X ×X.
Theorem 3. Suppose that X is a GO-space that has a dense S\delta-diagonal. If R \cup L is countable, then X* and L(X) have dense S\delta-diagonals.
Date received: July 15, 2001
Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagx-37.