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In this paper we introduce and study a new class of sets, namely γ−countably paracompact sets. We characterize γ−countably paracompact sets and we study some of its basic properties. We obtain that this class of sets is weaker than α−countably paracompact sets and stronger than β−countably paracompact sets.

In this paper, with the way-below relation ≪ and the way-above relation ≪ d , we give a poset-valued insertion theorem by a pair of semi-continuous maps. Moreover, we show a poset-valued insertion theorem by a continuous map as follows: Let X be a paracompact Hausdorff space, P a bi-bounded complete, bicontinuous, pathwise connected, topological poset. For each upper semi-continuous map f : X → P with a lower bound and each lower semi-continuous map g : X → P , if 〈 f , g 〉 has interpolated points pointwise, there exists a continuous map h : X → P such that f ≪ d h ≪ g . A generalized Dowker–Katětov's insertion theorem, by using the way-below and -above relations on bicontinuous posets P , is also given. [ABSTRACT FROM AUTHOR]

We construct a locally compact, monotonically normal, screenable, hereditarily paracompact space Y on P × ω in ZFC. The space Y is also a D-space in ZFC. The space Y is shown to be a locally compact, monotonically normal Dowker space and is not a D-space under ZF+AD and other models of set-theory. [ABSTRACT FROM AUTHOR]

For an ordered topological vector space Y and a , b ∈ Y , we write a ≪ b if b − a is an interior point of the positive cone. Modifying the earlier results of Borwein–Théra, in this paper, ≪ is extended over Y • • = Y ∪ { ∞ } ∪ { − ∞ } and a natural topology on Y • • is introduced. For a topological space X , and a non-trivial separable ordered topological vector space Y with an interior point of the positive cone, we show the following: X is normal and countably paracompact if and only if for every lower semi-continuous map f : X → Y • • and every upper semi-continuous map g : X → Y • • with g ≪ f , there exists a continuous map h : X → Y such that g ≪ h ≪ f . [ABSTRACT FROM AUTHOR]

It is shown that if $$X$$ contains a closed uncountable discrete subspace, then the Tychonoff plank embeds in the hyperspace $$CL(X)$$ of the non-empty closed subsets of $$X$$ with the Fell topology $$\tau _F$$ as a closed subspace. As a consequence, a plethora of properties is proved to be equivalent to normality and metrizability, respectively, of $$(CL(X),\tau _F)$$ . Countable paracompactness, pseudonormality and other weak normality properties of the Fell topology are also characterized. [ABSTRACT FROM AUTHOR]

Let X be a set and ℒ a lattice of subsets of X such that ∅, X∈ℒ. A(ℒ) is the algebra generated by ℒ, M(ℒ) the set of nontrivial, finite, normegative, finitely additive measures on A(ℒ) and I(ℒ) those elements of M(ℒ) which just assume the values zero and one. Various subsets of M(ℒ) and I(ℒ) are included which display smoothness and regularity properties.

Associated with a 0−1 measure μ∈I(ℒ) where ℒ is a lattice of subsets of X are outer measures μ′ and μ˜; associated with a σ-smooth 0−1 measure μ∈Iσ(ℒ) is an outer measure μ″ or with μ∈Iσ(ℒ′), ℒ′ being the complementary lattice, another outer measure μ˜˜. These outer measures and their associated measurable sets are used to establish separation properties on ℒ and regularity and σ-smoothness of μ. Separation properties between two lattices ℒ1 and ℒ2, ℒ1⫅ℒ2, are similarly investigated. Notions of strongly σ-smooth and slightly regular measures are also used.

Using a previously unexplored technique we construct a (transparently) hereditarily normal, collectionwise normal, screenable space Z on P × ω in ZFC. With more effort, the space Z is shown to be hereditarily paracompact and is hereditarily a D-space in ZFC. However, Z is easily shown to be a collectionwise normal, screenable Dowker space and is not a D-space in ZF+AD. We also observe that Z is a monotonically normal Dowker space in ZF+AD. It is known that such do not exist in ZFC. If one wants to avoid large cardinals there is a model due to S. Shelah [16] , equiconsistent with ZF , where Z is a CWN screenable Dowker space. [ABSTRACT FROM AUTHOR]

Abstract: A space X is said to have property (USC) (resp. (LSC)) if whenever is a sequence of upper (resp. lower) semicontinuous functions from X into the closed unit interval converging pointwise to the constant function 0 with the value 0, there is a sequence of continuous functions from X into such that () and converges pointwise to 0. In this paper, we study spaces having these properties and related ones. In particular, we show that (a) for a subset X of the real line, X has property (USC) if and only if it is a σ-set; (b) if X is a space of non-measurable cardinal and has property (LSC), then it is discrete. Our research comes of Scheepers'' conjecture on properties and wQN. [Copyright &y& Elsevier]

Abstract: Let be integers such that and . We show that there is a first countable, separable, -compact, countably paracompact and normal space with , and and point out some interesting questions concerning the gap between the dimensions of paracompact spaces. [Copyright &y& Elsevier]

A sufficient condition for the strict insertion of a continuous function between two comparable upper and lower semicontinuous functions on a normal space is given. Among immediate corollaries are the classical insertion theorems of Michael and Dowker. Our insertion lemma also provides purely topological proofs of some standard results on closed subsets of normal spaces which normally depend upon uniform convergence of series of continuous functions. We also establish a Tietze-type extension theorem characterizing closed G δ -sets in a normal space. [ABSTRACT FROM AUTHOR]

We show that a monotonically normal space X is paracompact if and only if for every increasing open cover { U α : α < κ} of X, there is a closed cover { F nα : n < ω, α < κ} of X such that F nα ⊂ U α for n < ω, α < κ and F nα ⊂ F nβ if α ≦ β. [ABSTRACT FROM AUTHOR]

Abstract: Let A and B be subspaces of an ordinal. It is proved that the product is countably paracompact if and only if it is rectangular. Before this main result, we discuss several covering properties of products with one ordinal factor. In particular, for every paracompact space X, it is proved that the product is paracompact if so is A. [Copyright &y& Elsevier]

Abstract: For an ordinal α, denotes the collection of all nonempty closed sets of α with the Vietoris topology and denotes the collection of all nonempty compact sets of α with the subspace topology of . It is well known that is normal iff . In this paper, we will prove that for every nonzero-ordinal α: [(1)] is countably paracompact iff . [(2)] is countably paracompact. [(3)] is normal iff, if cfα is uncountable, then . In (3), we use elementary submodel techniques. [Copyright &y& Elsevier]

Abstract: We present an example of a σ-product that is not countably paracompact but all of whose finite subproducts are countably paracompact. This example also shows that countable paracompactness of a σ-product may depend on the choice of base point. We also show that normal non-trivial σ-products are countably paracompact, improving a result of Chiba. Finally we give a new proof that σ-products of ordinals at base point 0 are κ-normal and strongly zero-dimensional. [Copyright &y& Elsevier]

Let A and B be two subspaces of an ordinal. It is proved that the product A × B is countably paracompact if and only if for every regular uncountable cardinal κ and for every closed rectangle A ′ × B ′ in A × B , either A ′ or B ′ contains a closed discrete subset of size κ whenever so does A ′ × B ′. This characterization of the countably paracompact products of ordinals is much simpler than the previous one given in 1992. [ABSTRACT FROM AUTHOR]

In 1997 Buzyakova [1] proved that for a Tychonoff, pseudocompact space X , X × (| β X | + + 1) can be condensed onto a normal space if and only if X can be condensed onto a compact, Hausdorff space. This elegant result shows the equivalence between condensation to a compact space and condensation of the product of a space with a compact space to a normal space. We prove the following result: If X is Tychonoff and X × (κ + 1) can be condensed onto a normal space, then X can be condensed onto a normal, countably paracompact space, where κ = (2 2 | X | ) +. [ABSTRACT FROM AUTHOR]

Abstract: We correct the proof of Theorem 8 in “Normality and countable paracompactness of hyperspaces of ordinals” [Topology Appl. 154 (2007) 358–362]. [ABSTRACT FROM AUTHOR]