Extension of point-finite partitions of unity

Haruto Ohta, and Kaori Yamazaki. "Extension of point-finite partitions of unity." Fundamenta Mathematicae 191.3 (2006): 187-199. <http://eudml.org/doc/286561>.

@article{HarutoOhta2006,
abstract = {A subspace A of a topological space X is said to be $P^\{γ\}$-embedded ($P^\{γ\}$(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 $P^\{γ\}$(point-finite)-embedded in X iff it is $P^\{γ\}$-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 $P^\{γ\}$(point-finite)-embedded in X × [0,1] iff A × Y is $P^\{γ\}$(point-finite)-embedded in X × Y for every compact Hausdorff space Y with w(Y) ≤ γ iff A is $P^\{γ\}$-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 $P^\{γ\}$(point-finite)-embedded in X × [0,1] is stronger than that of A being $P^\{γ\}$(point-finite)-embedded in X.},
author = {Haruto Ohta, Kaori Yamazaki},
journal = {Fundamenta Mathematicae},
keywords = {partition of unity; extension; product; analytic set; -embedding; -embedding; -embedding; -complete; AR},
language = {eng},
number = {3},
pages = {187-199},
title = {Extension of point-finite partitions of unity},
url = {http://eudml.org/doc/286561},
volume = {191},
year = {2006},
}

TY - JOUR
AU - Haruto Ohta
AU - Kaori Yamazaki
TI - Extension of point-finite partitions of unity
JO - Fundamenta Mathematicae
PY - 2006
VL - 191
IS - 3
SP - 187
EP - 199
AB - A subspace A of a topological space X is said to be $P^{γ}$-embedded ($P^{γ}$(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 $P^{γ}$(point-finite)-embedded in X iff it is $P^{γ}$-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 $P^{γ}$(point-finite)-embedded in X × [0,1] iff A × Y is $P^{γ}$(point-finite)-embedded in X × Y for every compact Hausdorff space Y with w(Y) ≤ γ iff A is $P^{γ}$-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 $P^{γ}$(point-finite)-embedded in X × [0,1] is stronger than that of A being $P^{γ}$(point-finite)-embedded in X.
LA - eng
KW - partition of unity; extension; product; analytic set; -embedding; -embedding; -embedding; -complete; AR
UR - http://eudml.org/doc/286561
ER -