Mary Ellen Rudin

Martin's Axiom

Publisher Summary This chapter presents various theorems to generalize when one should expect Martins axiom (MA) to be useful. Some of the theorems discussed in the chapter are if ZFC (Zermelo–Fraenkel set theory with the axiom of choice) is consistent, ZFC + MA + ¬ continuum hypothesis (CH) is consistent; if X is a ccc compact Hausdorff space, X is not the union of less than 2 ω nowhere dense sets; there is no Souslin tree; the product of any family of ccc spaces is ccc; and every Aronszajn tree is the union of countably many antichains. In topology MA + ¬ CH can be used to construct a variety of normal but not collectionwise normal spaces. MA can be used to deny the existence of certain pathologies in countable antichain condition (ccc) spaces.

Nikiel's conjecture

Abstract Nikiels conjecture that a topological space is the continuous image of a compact, linearly ordered space if and only if it is compact and monotonically normal is proved.

The box product of countably many compact metric spaces

Abstract It is proved that the continuum hypothesis implies that the box product of countably many σ -compact, locally compact, metric spaces is paracompact.

A normal screenable non-paracompact space

Abstract We construct a normal screenable non-paracompact spaces using ◊ ++ , which is a consequence of V = L .

Normality of product spaces and Morita's conjectures

Abstract It is well-known that Z is a perfectly normal space (normal P-space) if and only if X×Z is perfectly normal (normal) for every metric space X. Conversely, denote by Q (resp.N) the class of all spaces X whose products X×Z with all perfectly normal spaces (all normal P-spaces) Z are normal. It is natural to ask whether Q and N necessarily coincide with the class M of metrizable spaces. Clearly, M ⊂ N ⊂ Q . We prove that first countable members of Q are metrizable and that under V=L the classes M and N coincide, thus giving a consistency proof of Moritas conjecture. On the other hand, even though Q contains non-metrizable members, it is quite close to M: the class Q is countably productive and hereditary, and all members X of Q are stratifiable and satisfy c(X)=l(X)=w(X). In particular, locally Lindelof or locally Souslin or locally p-spaces in Q are metrizable. The above results immediately lead to the consistency proof of another Moritas conjecture, stating that X is a metrizable σ-locally compact space if and only if X×Y is normal for every normal countably paracompact space Y. No additional set-theoretic assumptions are necessary if X is first countable. In our investigation, an important role is played by the famous Bing examples of normal, non-collectionwise normal spaces. Answering Dennis Burkes question, we prove that products of two Bing-type examples are always non-normal.

Set-theoretic constructions of nonshrinking open covers

Abstract A family { M α | αϵA } is a shrinking of a cover { O α | αϵA } of a topological space if { M α | αϵA } also covers and M α ⊂ O α for all αϵA . ◊ ++ implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking. ◊ implies that there is a P-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in [17] to show that any space which has a normal product with every P-space is metrizable.

Monotone normality and compactness

Abstract A locally compact monotonically normal space having no compactification which is monotonically normal is given as well as a consistent example of a compact K 1 -space which is not K 0 .

Countable box products of ordinals

The countable box product of ordinals is examined in the paper for normality and paracompactness. The continuum hypothesis is used to prove that the box product of countably many a-compact ordinals is paracompact and that the box product of another class of ordinals is normal. A third class trivially has a nonnormal product. Because I have found a countable box product of ordinals useful in the past [1], this class of spaces particularly interests me. The purpose of this paper is to tell what I know about which of these spaces is paracompact or normal. In [2] I prove that the continuum hypothesis implies the box product of countably many a-compact, locally compact, metric spaces is paracompact. I prove here that the continuum hypothesis implies the box product of countably many a-compact ordinals is paracompact (Theorem 1) and the box product of another class of ordinals is normal (Theorem 2). The proof of Theorems 1 and 2 is a quite messy join of the techniques of [1] and [2] which raises some doubt in my mind as to whether these theorems are worth proving. Because I care, because I think these spaces are set theoretically interesting and topologically useful, because I think these theorems are best possible, the theorems are worth the mess to me. A. If {XXAOA)\ is a family of topological spaces, a box in HIAeA XA is a set TheA UA where each UA is open in XA. The box product of {XAIA A iS HAA XA topologized by using the set of all boxes in it as a basis. Throughout the paper the following notation is used. An ordinal a is the set of all ordinals less than a and a is topologized by the interval topology. The statement that a is a cardinal means that a is an ordinal and no smaller ordinal has the same cardinality as a. The notation IAnA BAA is used to mean the ordinary Cartesian product of the f,3s and never the cardinal or ordinal arithmetic product. Similarly a#9 means the set of all functions from ,B into a rather than an arithmetic operation. If a is an ordinal, let cf(a) denote the cofinality of a; that is cf(a) is the smallest ordinal 8 such that there is a subset A of a, order isomorphic with 8, such that /3 < a implies there is a y El A with /3 < y. Observe that a is a a-compact ordinal if and only if a is compact or cf(a) =w. Received by the editors December 10, 1971 and, in revised form, October 1, 1972. AMS (MOS) subject classiflcations (1970). Primary 54B10, 54A25, 54D15, 54D20, 54D30, 02K25.

S AND L SPACES

Publisher Summary This chapter discusses the S and L spaces. An S space is a T3 topological space that is hereditarily separable but not Lindelof. An L space is a T3 topological space that is hereditarily Lindelof but not separable. There is no obvious duality between separability and Lindelofness. Spaces such as the Cantor tree or the tangent disk space are separable but not Lindelof. However, these spaces all have uncountable discrete subspaces and are, thus, neither hereditarily Lindelof nor hereditarily separable. The S and L problems are a part of a vital movement in general topology to think of topological properties as cardinal functions [Ju1], [Ru3]. For a cardinal α, a space can be defined to to be α-separable if α is the minimal cardinality of a dense set and α-Lindelof if α is the minimal cardinal such that every open cover has a subscover of that cardinality.

Compact, separable, linearly ordered spaces

Abstract A proof that a compact, separable, zero-dimensional, monotonically normal space is always a continuous image of a compact linearly ordered space is given.