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1

Diagonal conditions in ordered spaces

100%

Bennett H. R., Lutzer D. J.

Fundamenta Mathematicae

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1997

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tom 153

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nr 2

99-123

EN

For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each $T ⊂ {X^2} - Δ(X)$ with |T| = κ, there is an open neighborhood W of Δ(X) such that |T - W| = κ. If $ω_1 ∈ D(X)$ then we say that X has a small diagonal, and if every regular uncountable κ ≤ |X| belongs to D(X) then we say that X has an H-diagonal. In this paper we investigate the interplay between D(X) and topological properties of X in the category of generalized ordered spaces. We obtain cardinal invariant theorems and metrization theorems for such spaces, proving, for example, that a Lindelöf linearly ordered space with a small diagonal is metrizable. We give examples showing that our results are the sharpest possible, e.g., that there is a first countable, perfect, paracompact Čech-complete linearly ordered space with an H-diagonal that is not metrizable. Our example shows that a recent CH-result of Juhász and Szentmiklóssy on metrizability of compact Hausdorff spaces with small diagonals cannot be generalized beyond the class of locally compact spaces. We present examples showing the interplay of the above diagonal conditions with set theory in a natural extension of the Michael line construction.

2

Ordered spaces with special bases

100%

Bennett H. R., Lutzer D. J.

Fundamenta Mathematicae

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1998

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tom 158

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nr 3

289-299

EN

We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a $G_δ$-diagonal, that a linearly ordered space has a point-countable base if and only if it is first-countable and has an ω-in-ω base, and that metrizability in a generalized ordered space is equivalent to the existence of an OIF base and to the existence of a sharp base. We give examples showing that these are the best possible results.

3

Dugundji extenders and retracts on generalized ordered spaces

100%

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

2 Bennett H. R.

2 Lutzer D. J.

1 Gruenhage G.

1 Hattori Y.

1 Ohta H.

2 1998

1 1997

Diagonal conditions in ordered spaces

Ordered spaces with special bases

Dugundji extenders and retracts on generalized ordered spaces