William G. Fleissner

Strong zero-dimensionality of products of ordinals

Abstract We show that the product of finitely many subspaces of ordinals is strongly zero-dimensional. In contrast, for each natural number n, there is a subspace of (ω+1)× c of dimension n.

Collectionwise Hausdorff: incompactness at singulars

Abstract Under set theoretic hypotheses, we construct a λ-collectionwise Hausdorff not λ + - collectionwise Hausdorff space of character c for certain singular cardinals λ. For example if V = L , and cf(λ) is not weakly compact, or if there are no inner models with large cardinals, λ is singular strong limit, and cf(λ) is the successor of a singular strong limit. Moreover, after forcing collapsing c to ω these spaces retain their properties; thus we obtain first countable examples.

Killing normality with a Cohen real

Abstract We describe a machine that inputs a normal space X and outputs a normal superspace T such that T becomes nonnormal after adding one Cohen real if and only if X is a Dowker space. A similar construction applied to Rudins box product Dowker space yields a collectionwise normal space that becomes nonnormal after the addition of one Cohen real.

Ordered spaces all of whose continuous images are normal

This is the published version, also available here: http://dx.doi.org/10.1090/S0002-9939-1989-0973846-4. First published in Proc. AMS. in 1989, published by the American Mathematical Society.

Not realcompact images of not Lindelöf spaces

Abstract X has not a realcompact image if X is not Lindelof and at least one of the following: X has Lindelof degree at most 2 ω , X is countability tight, X maps continuously onto X 2 . If X has an uncountable closed discrete subset, then X has a one-to-one not realcompact continuous image. If X is not Lindelof, then X , the sum of the finite powers of X , has a not realcompact continuous image; hence L ( X ) = pq ( pq ) = t ( C p ( X )).

Normal measure axiom and Balogh's theorems

Abstract We present an axiom and combinatorics approach to recent results of Balogh. Specifically we prove that the Normal Measure Axiom implies that normal locally compact spaces are collectionwise normal and that countably paracompact, locally compact spaces are expandible. We present a proof of Prikrys theorem that adding supercompact many random reals forces the Normal Measure Axiom.

Normal subspaces of products of finitely many ordinals

Let X be a subspace of the product of finitely many ordinals. If X is normal, then X is strongly zero-dimensional, collectionwise normal, and shrinking. The proof uses (κ 1 ,...,κ n )-stationary sets.

Which spaces have a coarser connected Hausdorff topology

Abstract We present some answers to the title. For example, if K is compact, zero-dimensional and D is discrete, then K⊕D has a coarser connected topology iff w(K)⩽2|D|. Similar theorems hold for ordinal spaces and spaces K⊕D where K is compact, not necessarily zero-dimensional. Every infinite cardinal has a coarser connected Hausdorff topology; so do Kunen lines, Ostaszewski spaces, and Ψ-spaces; but spaces X with X⊂βω and |βω⧹X| c do not. The statement “every locally countable, locally compact extension of ω with cardinality ω1 has a coarser connected topology” is consistent with and independent of ZFC. If X is a Hausdorff space and w(X)⩽2κ, then X can be embedded in a Hausdorff space of density κ.

Metacompact subspaces of products of ordinals

This is the published version, also available here: http://dx.doi.org/10.1090/S0002-9939-01-06026-9. First published in Proc. Amer. Math. Soc. in 2001, published by the American Mathematical Society.

Remainders of normal spaces

Abstract Fleissner, W., J. Kulesza and R. Levy, Remainders of normal spaces, Topology and its Applications 49 (1993) 167-174. If X is totally compact, then X is the Stone-tech remainder of a normal space. A partial converse: if X is first countable and the Stone-tech remainder of a normal space, then X is locally compact. Every metric space, but not every first countable space, is the remainder of a normal space. For countable spaces, or even countable spaces which are locally compact except at one point, there are examples, but few theorems. We show that a construction of Porter and Woods applies to certain examples only if b = b. We investigate the property that all normal images are compact, and show that large products minus small subsets have this property. Keywords: Remainder, normal, locally compact, countable type, Stone-tech, Z-product, count- able spaces.