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On Nagata's star-index and covering dimension of spaces
by
Yasunao Hattori
Shimane University
Coauthors: Kaori Hashimoto (Shimane University)
In 1979, Bruijning and Nagata introduced a function \Deltak(X) for a topological space X and a natural number k to give an interesting characterization of the covering dimension which is motivated by the classsical theorem of Pontrjagin and Schnirelmann. The function \Deltak(X) is defined as the least natural number m such that for every cozero-set covering U of X with the cardinality k there exists a cozero-set covering V of X with the cardinality not more than m such that V is a delta-refinement of U. Later, Nagata (1984) introduced a function star-index *k(X) for a normal space X and a natural number k by modifying \Deltak(X); i.e., *k(X) is the least natural number m such that for every open covering U of the cardinality k there exists an open covering V of X with the cardinality is not more than m such that V is a star-refinement of U. Then he asked if the value of *k(X) can be determined. I shall present a positive answer the question. I shall also present a characterization of the covering dimension in terms of the star-index originated by Nagata.
Date received: February 1, 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 # cagh-29.