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2006 | 191 | 3 | 187-199
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Extension of point-finite partitions of unity

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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.
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  • Faculty of Education, Shizuoka University, Ohya, Shizuoka 422-8529, Japan
  • Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm191-3-1
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