Dennis K. Burke

A nondevelopable locally compact hausdorff space with a Gδ-diagonal

Abstract An example is constructed of a locally compact T 2 space which has a G δ -diagonal but is not developable. This answers negatively the question of whether every p-space with a G δ -diagonal is developable. A perfect map is constructed from this space onto a space which does not have a G δ -diagonal: thus showing that a G δ -diagonal need not be preserved under a perfect map.

Subspaces of the Sorgenfrey Line

Abstract We study three problems which involve the nature of subspaces of the Sorgenfrey Line S . It is shown that no integer power of an uncountable subspace of S can be embedded in a smaller power of S . We review the known results about the existence of uncountable X ⊆ S where X2 is Lindelof. These results about Lindelof powers are quite set-theoretic. A descriptive characterization is given of those subspaces of S which are homeomorphic to S . We show that a nonempty subspace Z ⊆ S is homeomorphic to S if and only if Z is dense-in-itself and is both Fσ and Gδ in S .

ON COUNTABLY COMPACT EXTENSIONS OF NORMAL LOCALLY COMPACT M-SPACES

Publisher Summary This chapter discusses the compact extensions of normal locally compact M-spaces. All spaces are assumed to be completely regular and T1. A space X is an M-space [Mo1] if there exists a metrizable space T and a mapping f: X → T, where f is a quasi-perfect map. A space Z is a countably-compactification of a space X if Z is countably compact, X is a dense subset of Z, and every closed countably compact subset of X is also closed in Z. If a space X is countably-compactifiable, then X has a countably-compactification Z such that X ⊂ Z ⊂ βX.

Functional characterizations of certain

Abstract In this paper we prove that p-spaces and Cech-complete spaces with G δ -diagonals can be characterized by a familiar function extension property.

p

Abstract In this paper we answer questions of van Douwen and Pfeffer by showing that the spaces S , S 2 , S 3 , and T , T 2 , T 3 ,... are topologically distinct, where S is the Sorgenfrey line and T is the set of irrational points in S . We obtain analogous results for the Michael line M and present related examples.

-spaces

A new characterization of spaces having a point-countable basis is obtained. This characterization is used in giving a simpler proof of a recent theorem of Filippov.

On powers of certain lines

For disjoint subsets A,C of [0, 1] the Michael space M(A,C) = A ∪ C has the topology obtained by isolating the points in C and letting the points in A retain the neighborhoods inherited from [0, 1]. We study normality of the product of Michael spaces with complete metric spaces. There is a ZFC example of a Lindelof Michael space M(A,C), of minimal weight א1, with M(A,C) × B(א0) Lindelof but with M(A,C) × B(א1) not normal. (B(אα) denotes the countable product of a discrete space of cardinality אα.) If M(A) denotes M(A, [0, 1]rA), the normality of M(A)×B(א0) implies the normality of M(A)× S for any complete metric space S (of arbitrary weight). However, the statement “M(A,C)×B(א1) normal implies M(A,C)×B(א2) normal” is axiom sensitive.

On a theorem of V. V. Filippov

Abstract We address the following question: “Must every w Δ-space with a G δ -diagonal be developable?” Consistently, the answer is “no.” Example . Assume CH. There is a zero-dimensional, scattered, locally compact, w Δ-space with a G δ -diagonal which is not developable. For normal, locally compact spaces (or slightly weaker), the answer is “yes”. Theorem . If X is ω- s CWH, locally Lindelof, w Δ-space with a G δ -diagonal, then X is developable.

Products of Michael spaces and completely metrizable spaces

Abstract A class of Baire spaces, which contains many known examples and variations thereof, is described and it is shown that no space in this class contains a dense metrizable G δ -subspace. This gives a class of semi-metrizable spaces which are not σ-spaces. We discuss the existence of Lindelof semi-metrizable spaces which are not σ-spaces. This is of interest since the only known examples require the use of CH.

The wΔ-space problem

Abstract The main result shows that the class of spaces with a δθ-base is invariant under perfect mappings. By using related techniques it is also shown that the class of weakly θ-refinable spaces is preserved by perfect images.