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Extension of point-finite partitions of unity

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Ohta H., Yamazaki K.

Fundamenta Mathematicae

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2006

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tom 191

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nr 3

187-199

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.

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Does C* -embedding imply C*-embedding in the realm of products with a non-discrete metric factor?

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Gutev V. G., Ohta H.

Fundamenta Mathematicae

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2000

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tom 163

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nr 3

241-265

EN

The above question was raised by Teodor Przymusiński in May, 1983, in an unpublished manuscript of his. Later on, it was recognized by Takao Hoshina as a question that is of fundamental importance in the theory of rectangular normality. The present paper provides a complete affirmative solution. The technique developed for the purpose allows one to answer also another question of Przymusiński's.

3

Dugundji extenders and retracts on generalized ordered spaces

81%

Gruenhage G., Hattori Y., Ohta H.

Fundamenta Mathematicae

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1998

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tom 158

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nr 2

147-164

EN

For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an $L_{ch}$-extender (resp. $L_{cch}$-extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X\A; (iii) there is a continuous $L_{ch}$-extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology and the pointwise convergence topology, for each space Y; (iv) A × Y is C*-embedded in X × Y for each space Y; (v) there is a continuous linear extender $φ:C*_{k}(A) → C_{p}(X)$; (vi) there is an $L_{ch}$-extender φ:C(A) → C(X); and (vii) there is an $L_{cch}$-extender φ:C(A) → C(X). We prove that these conditions are related as follows: (i)⇒(ii)⇔(iii)⇔(iv)⇔(v)⇒(vi)⇒(vii). If A is paracompact and the cellularity of A is nonmeasurable, then (ii)-(vii) are equivalent. If there is no connected subset of X which meets distinct convex components of A, then (ii) implies (i). We show that van Douwen's example of a separable GO-space satisfies none of the above conditions, which answers questions of Heath-Lutzer [9], van Douwen [1] and Hattori [8].

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Ohta H.

1 Gruenhage G.

1 Gutev V. G.

1 Hattori Y.

1 Yamazaki K.

1 2006

1 2000

1 1998

Extension of point-finite partitions of unity

Does C* -embedding imply C*-embedding in the realm of products with a non-discrete metric factor?

Dugundji extenders and retracts on generalized ordered spaces