Relative Semi-metrics
by
Gary Grabner
Slippery Rock University
Coauthors: Elise Grabner, Kazumi Miyazaki and Jamal Tartir

We study several properites of relative semi-metrizable type based on work of Arhangel'skii and Gorienko in [1]. We characterize of these properites using relative versions of first countability and semi-stratifiable and give examples distingushing them.

Theorem 1 A subspace Y is strongly semi-stratifiable in X and strongly first countable in X if and only if there is a symmetric on (Y,X) properly defining Y in X and satisfying condition O).

Theorem 2 Y is 03) - semi-metizable in X iff Y is 2- first countable in X and semi-stratifiable in X.

Theorem 3 If Y is developable in X then there is a 02)- semi-metric on (Y,X).

[1] A. Arhangel'skii and I. Gordienko, "Relative symmetrizability and metrizability" Comment. Math. Univ. Carol., Vol 37 (1996), 757-774

Date received: February 20, 2006