Perfect Preimages Research Articles

The normality of products under perfect preimages

A proof of the following theorem is given, answering an open problem attributed to Kunen: suppose that T is compact and that Y is the image of X under a perfect map, X is normal, and Y×T is normal. Then X×T is normal.

PFA(S) and countable tightness

Todorcevic introduced the forcing axiom PFA(S) and established many consequences. We contribute to this project. In particular, we consider status under PFA(S) of two important consequences of PFA concerning spaces of countable tightness. We prove that the existence of a Souslin tree does not imply the existence of a compact non-sequential space of countable tightness. We contrast this with M.E. Rudin's result that the existence of a Souslin tree does imply the existence of an S-space (and the later improvement by Dahrough to a compact S-space). On the other hand, PFA(S) implies there is a first-countable perfect pre-image of ω1 that contains no copies of ω1.

On the Hereditary Paracompactness of Locally Compact, Hereditarily Normal Spaces

AbstractWe establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space that does not include a perfect pre-image of ω1 is hereditarily paracompact.

Formula omitted] and locally compact normal spaces

We examine locally compact normal spaces in models of form PFA(S)[S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of ω1 and in which all separable closed subspaces are Lindelöf.

Applications of some strong set-theoretic axioms to locally compact T5and hereditarily scwH spaces

Under some very strong set-theoretic hypotheses, hereditarily normal spaces (also referred to as T5 spaces) that are locally compact and hereditarily collec- tionwise Hausdor can have a highly simplied structure. This paper gives a structure theorem (Theorem 1) that applies to all such !1-compact spaces and another (Theorem 4) to all such spaces of Lindelof number @1. It also introduces an axiom (Axiom F) on crowding of functions, with consequences (Theorem 3) for the crowding of countably compact subspaces in certain continuous preimages of !1. It also exposes (Theorem 2) the ne structure of perfect preimages of !1 which are T5 and hereditarily collectionwise In these theorems, \T5 and hereditarily collectionwise Hausdor is weakened to \hereditarily strongly collectionwise Hausdor. Corollaries include the consistency, modulo large cardinals, of every hereditarily strongly collectionwise Hausdor manifold of dimension > 1 being metrizable. The concept of an alignment plays an important role in formulating several of the structure theorems.

On perfect pre-images of ω1

We show that PFA implies that every perfect pre-image of ω1 of countable tightness contains a closed copy of ω1.

Shrinking in perfect preimages of shrinking spaces

Every normal perfect preimage of a shrinking space is ω 1-shrinking. Every normal perfect preimage of a monotonically normal space is shrinking.

Measure-compact spaces

The property measure-compact fits between the Lindelöf and realcompact properties, i.e., Lindelöf implies measure-compact implies realcompact. Much of the research on measure-compact spaces centers on two themes: one, what properties measure-compactness shares with Lindelöf and realcompactness and two, the relationships between measure-compactness and covering properties. The results in this paper can be grouped into four parts illustrating these two themes. First, we show that F σ subsets of measure-compact spaces are measure-compact (Theorem 2.9). This theorem improves results of Okada and Okasaki and answers a question of Wheeler. Second, we prove that perfect preimages of measure-compact spaces are measure-compact (Theorem 2.18). The results analogous to these are true for both Lindelöf and realcompact spaces. Third, we examine the relationship between measure-compactness and covering properties. Fourth, we present some examples. Without any extra axioms of set theory, we present a locally compact, realcompact, not measure-compact space; a locally compact, measure-compact, not paracompact space; a normal, metacompact, measure-compact, not paracompact space; and a nonmeasure-compact space which is the union of a measure-compact space and a σ-compact space. The first two examples answer questions of Kirk and Wheeler, respectively. Spaces in the prior literature with the properties of the third and fourth examples used extra set-theoretic axioms.

Preimages of Spaces Having a Base of Countable Order

The main theorem characterizes the quasi‐perfect and perfect preimages of T1‐spaces having a base of countable order. The preimages of T1‐spaces having a base of countable order are connected with the preimages of T1‐spaces having a primitive base under the same type of mappings.

Wallman prebases

Let X be completely regular and Hausdorff. We define the notion of a Wallman prebase for X in a manner analogous to that of the Wallman base used in the construction of a Wallman compactification of X. Corresponding to each Wallman prebase of X is a closed completely regular Hausdorff image of X. Conversely, every closed completely regualr Hausdorff image Y of X gives rise to a number of Wallman prebases of X, one for each Wallman base of Y. A natural way of identifying “equivalent” Wallman prebases yields a 1-1 correspondence between prebases and closed images. Restrictions on the Wallman prebase yield images and maps satisfying stronger conditions. In particular, a space is the perfect preimage of a rimcompact space if and only if it has a Wallman prebase satisfying certain conditions.

Perfect pre-images of ω 1 and the PFA

Assuming the Proper Forcing Axiom, I give a widely applicable condition under which a pre-image of ω 1 includes a homeomorphic image of ω 1. This makes it possible to strengthen a number of results of Z. Balogh and P.J. Nyikos on non-metrizable manifolds and related spaces.

On closed images of perfect preimages of orthocompact developable spaces

On closed images of perfect preimages of orthocompact developable spaces

Even Covering Properties and Somewhat Normal Spaces

AbstractA topological space X is said to be somewhat normal provided that for each open cover is a normal cover of X. We show that a completely regular somewhat normal space need not be normal, thereby answering a question of W. M. Fleischman. We note that a collectionwise normal somewhat normal space need not be almost 2-fully normal, as had previously been asserted, and that neither the perfect image nor the perfect preimage of a somewhat normal space has to be somewhat normal.

Perfect Pre-Images of Collectionwise Normal Spaces

Perfect Pre-Images of Collectionwise Normal Spaces

Perfect pre-images of collectionwise normal spaces

We show that the following are true for perfect maps: (1) collectionwise normality with respect to compact (Lindelöf) sets is preserved inversely; (2) collectionwise normality with respect to countably compact ( ω 1 {\omega _1} -compact) sets is preserved inversely if and only if the domain space is normal with respect to countably compact ( ω 1 {\omega _1} -compact) sets; and (3) if P P is any property such that (i) P P is preserved by perfect maps, (ii) the free union of spaces satisfying P P also satisfies P P , (iii) P P is closed hereditary, and (iv) P P plus collectionwise normality implies countable metacompactness, then collectionwise normality with respect to closed P P -sets is preserved inversely if the domain space is normal with respect to closed P P -sets. Examples of such a property P P are paracompactness, submetacompactness, stratifiability and countable metacompactness.

The smallest basically disconnected preimage of a space

We show that for each space X, there exists a smallest basically disconnected perfect irreducible preimage Λ X. A corollary of the existence of Λ X is that each locally compact and basically disconnected space X has a smallest basically disconnected compactification.

Two examples on preimages of metric spaces

Examples are given to show that the product of a compact T1-space with a metric space need not be a k-space, and that the product of a countably compact Hausdorff space with a metric space need not be a quasi-k-space. These examples show that the separation assumptions cannot be omitted in the following two known results: Each Hausdorff perfect preimage of a k-space is a k-space. Each regular quasi-perfect preimage of a T1 quasi-kspace is a quasi-k-space. Some related results, perhaps of independent interest, are also obtained.