A subspace A of a topological space X is said to be -embedded ((point-finite)-embedded) in X if every (point-finite) partition of unity α on A with |α| ≤ γ extends to a (point-finite) partition of unity on X. The main results are: (Theorem A) A subspace A of X is (point-finite)-embedded in X iff it is -embedded and every countable intersection B of cozero-sets in X with B ∩ A = ∅ can be separated from A by a cozero-set in X. (Theorem B) The product A × [0,1] is (point-finite)-embedded in X × [0,1] iff A × Y is (point-finite)-embedded in X × Y for every compact Hausdorff space Y with w(Y) ≤ γ iff A is -embedded in X and every subset B of X obtained from zero-sets by means of the Suslin operation, with B ∩ A = ∅, can be separated from A by a cozero-set in X. These characterizations are used to answer certain questions of Dydak. In particular, it is shown that, assuming CH, the property of A × [0,1] to be (point-finite)-embedded in X × [0,1] is stronger than that of A being (point-finite)-embedded in X.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm191-3-1,
author = {Haruto Ohta and Kaori Yamazaki},
title = {Extension of point-finite partitions of unity},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {187-199},
zbl = {1145.54008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm191-3-1}
}
Haruto Ohta; Kaori Yamazaki. Extension of point-finite partitions of unity. Fundamenta Mathematicae, Tome 189 (2006) pp. 187-199. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm191-3-1/