Homogeneity of powers of spaces and the character

• Published in: Proceedings of the American Mathematical Society Vol. 133; no. 7; pp. 2165 - 2172
• Main Author: A. V. Arhangel'skii
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-07-2005
• Subjects: Cardinal points; Cardinality; Exact sciences and technology; First countable; General topology; Hausdorff spaces; Mathematical theorems; Mathematics; Numbers; Sciences and techniques of general use; Topological spaces; Topological theorems; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Power-homogeneous; first countable space; G_\tau $-tightness; tau$-diagonalizable space; point-countable type; tau$-twister; linearly ordered space; Topological space; Homogeneity; Hausdorff space; Homogeneous space; Tau diagonalizable space; Ordered space; Power
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-05-07774-9

A space is said to be power-homogeneous if some power of it is homogeneous. We prove that if a Hausdorff space X of point-countable type is power-homogeneous, then, for every infinite cardinal \tau , the set of points at which X has a base of cardinality not greater than \tau , is closed in X. Every power-homogeneous linearly ordered topological space also has this property. Further, if a linearly ordered space X of point-countable type is power-homogeneous, then X is first countable.