Extension of point-finite partitions of unity

• Autorzy: Haruto Ohta , Kaori Yamazaki
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• 54C55: Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
• 54C20: Extension of maps

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2006
• Tom: 191
• Numer: 3
• Strony: 187-199

Daty

wydano

2006

Twórcy

autor

Haruto Ohta

• Faculty of Education, Shizuoka University, Ohya, Shizuoka 422-8529, Japan

autor

Kaori Yamazaki

• Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan

Identyfikatory

DOI

10.4064/fm191-3-1