A dissertation presented to. the faculty of. In partial fulfillment. of the requirements for the degree. Doctor of Philosophy. Derrick D.

Transcription

1 Continuous Mappings and Some New Classes of Spaces A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Derrick D. Stover June Derrick D. Stover. All Rights Reserved.

2 This dissertation titled Continuous Mappings and Some New Classes of Spaces by DERRICK D. STOVER has been approved for the Department of Mathematics and the College of Arts and Sciences by Alexander V. Arhangel skii Distinguished Professor of Mathematics Benjamin M. Ogles Dean, College of Arts and Sciences 2

3 ABSTRACT STOVER, DERRICK D., Ph.D., June 2009, Mathematics Continuous Mappings and Some New Classes of Spaces (152 pp.) Director of dissertation: Alexander V. Arhangel skii The theory of continuous mappings is a crucial field of study in general topology. In this dissertation we expand on this vast area by considering continuous functions and some new classes of spaces, while also pursuing other natural considerations in this direction. A space X is said to be π-metrizable if it has a σ-discrete π-base. The behavior of π- metrizable spaces under certain types of mappings is studied. In particular we characterize stronglyd-separable spaces as those which are the image of a π-metrizable space under a perfect mapping. Each Tychonoff space can be represented as the image of a π-metrizable space under an open continuous mapping. A question posed by Arhangel'skii regarding if a π-metrizable topological group must be metrizable receives a negative answer. A space X is said to be coconnected if X >1 and for every connected subset C, X\C is connected. It is established that every coconnected space can be mapped onto a coconnected compactum by a continuous bijection. Also every coconnected compactum is the union of two linearly ordered continua intersecting only at end points. In particular every separable compact coconnected space is homeomorphic to the circumference. Every continuum that is cleavable over the class of coconnected spaces together with the class of LOTS embeds into a coconnected space. Thus cleavability of continua over the class of LOTS can be generalized to cleavability over coconnected spaces and their connected subsets. Cleavability over linearly ordered spaces has been an important direction of study. We establish that every locally connected, connected space cleavable over the class of LOTS is linearly 3

4 ordered. Every separable connected space cleavable over R condenses onto an interval. Finally it is shown that every space cleavable over the class of LOTS containing a converging sequence has a cut point. Approved: Alexander V. Arhangel skii Distinguished Professor of Mathematics 4

5 Preface The theory of continuous mappings is a crucial field of study in general topology. In this dissertation we expand on this vast area by considering continuous functions and some new classes of spaces, while also pursuing other natural considerations in this direction. By N we mean the set of all natural numbers. All topological spaces are assumed to satisfy the Tychonoff separation axiom unless otherwise specified. For a space X, a collection of nonempty open sets Θ, is called a π base if for every nonempty open set O, there exists U Θ such that U O. For a space X we denote by π(x) the least cardinal τ such that X has a π-base Γ with Γ = τ. A space X is said to be π metrizable if it has a σ-discrete π-base. The first chapter focuses on π-bases: especially π-metrizability. Metrizable spaces are among the nicest in general topology. So it is natural to consider modifications to some characterization of this class of spaces. In particular, it is a natural consideration to weaken the requirement of a σ-discrete base (see [14]) to that of a σ-discrete π-base. At this point we may ask How does this new class of spaces behave under mappings, subspaces, products, with the Baire property, etc? Some classical results of metrizable spaces have analogs when only a π-base is considered. In Section 1.1, every space with a σ-locally-finite π-base is shown to be π-metrizable. Also in this section, every pseudocompact space with a σ-point- 5

6 finite π-base is seen to have a σ-disjoint π-base. It is however, perhaps fair to say that the further one reads into chapter one, the less metrizable the spaces under consideration appear. One result headlining the first chapter is a characterization of those spaces which are the image of a π-metrizable space under a perfect (or just a closed) mapping as precisely those spaces which have a σ-discrete (in X) dense subset: such spaces I have called strongly-d-separable. The primary emphasis of chapter one really went in the direction of products. I initially considered two questions: If X Y is π-metrizable does this imply that both X and Y are so? Then the stronger question: Given a space X does there exist a space Y with X Y being π-metrizable? These questions were answered affirmatively by establishing Theorem 1.3.3, which is almost certainly the most important result in the entire first chapter. The theorem states that if X n has a discrete collection of κ open sets for all n N and π(y ), π(x n ) κ for all n, then Y ( {X n : n N}) is π-metrizable. This theorem also answers the question of if a π-metrizable topological group must be metrizable (negatively). Even a very weak consequence, that an uncountable product can be π-metrizable, is a highly nontrivial result. Though the biggest part of the results of the third section of chapter one are a consequence of the above mentioned theorem; I close with an attempt to move in the opposite direction, by proving that if X, to any power, is π-metrizable, then so 6

7 is the countable power of X. I am almost successful (see Theorem for details). The remaining sections of the first chapter take a very specialized form. In section four, conditions for a space to have a point-finite-π-base are considered: it is found to share a close relationship with being of the first Baire category. Then in the final section, I exhibit a special construction in which I define the π-boundary (see Section 1.5) and prove that every compactum is the π-boundary of some continuum. A space X is called coconnected if X is nondegenerate (more than one point) and the compliment to every connected set is connected. The concept of coconnectivity was first introduced by Dr. Arhangel skii in a special seminar on general topology at Ohio University in The question was posed that if a compact subset X of R 2 has the property that for every C X such that C is connected, X\C is also connected, then is it true that X is homeomorphic to S 1? (Of course we shall see much more is true.) At first glance coconnectivity is a very simple assumption. It has an almost playful feel to it. Yet coconnected spaces have a very prescribed form. Every coconnected space is the quotient space obtained by associating the end points of a linearly ordered space, possibly with a finer topology. Furthermore the least additional assumption, added to coconnectivity, tends also to be revealing. For example every locally connected, coconnected space is compact (as is every locally compact coconnected space). 7

8 After learning the form of coconnected spaces, I then considered coconnectivity into the direction of cleavability. A space X is said to be cleavable over a class of spaces Γ if for each A X there exists Y Γ and a continuous function f : X Y, such that f(a) = Y \f(x\a). A space X is said to be cleavable over a space Y if X is cleavable over the class of all subspaces of Y. These concepts were originally introduced in [7]. Much is known about cleavability over connected linearly ordered topological spaces (called LOTS). If the space X is assumed to be a continuum (a connected compactum not necessarily metrizable) and is cleavable over the class of LOTS then X is linearly ordered (see [12]). The results of Section 2.1 related coconnected spaces to these in an obvious way. I wished to determine if anything new was gained by considering also coconnected spaces. It is clear that we pick up all coconnected continua. As it turns out this our only gain. That is, if X is a continuum cleavable over the class over all coconnected spaces together with the class of linearly ordered spaces then X is linearly ordered or coconnected. For the next section the focus narrowed to simply considering cleavability over S 1. Every separable, connected, locally connected space cleavable over S 1 is shown to embed into S 1. The same is true of every connected, locally compact space cleavable over S 1. Of course these results remain true if R is put in place of S 1 : solving two previous open problems. The third and final chapter is dedicated to cleavability over LOTS. In the first 8

9 four sections I modestly generalize the previous cleavability results of Arhangelskii, Buzyakova and myself by weakening assumptions. If X is cleavable over the class of LOTS and X has property P then X is linearly ordered. This has been a traditional problem of cleavability. It has been established for P being that X is a continuum (see [12]), and it is an open problem if it is true if we assume only that the space is compact. I establish that it holds for X assumed to be connected and locally connected. Since dropping either condition produces a demonstrably false result, it is unlikely that any generalization of this fact will be possible. Many results on cleavability involve cleavability over a single space Y, rather than a class of spaces. Such premises allow for more specific results to be discovered, generally in the form of the space under consideration being very similar to Y. It has been shown that every compactum cleavable over R or over R N embeds into the respective space (see [2, 4] respectively). Buzyakova has shown that the same is true of the two arrows of Alexandroff (see [10]). Section 3.2 shows that every connected, separable space cleavable over R condenses onto R. This result also answers the question of whether or not each connected, separable space cleavable over R has a cut point (it in fact has many). Since every continuum cleavable over R is separable, this generalizes the result that every continuum cleavable over R is a segment (see [2]). Section 3.3 goes in a similar direction showing that every locally compact, 9

10 connected space cleavable over the Long Line is a subset of it. In the fourth section of the final chapter I search for a weaker result: the existance of cut points. It is a relatively unexplored area of cleavability and it was my hope that I could establish that every connected space cleavable over the class of LOTS, or even over R, has a cut point. I showed that this would happen provided that the space contains an infinite set with the property that no infinite subset of it is closed (in particular a converging sequence). However, for our present knowledge, every connected space cleavable over the class of LOTS (or at least over R) may indeed contain a converging sequence. In section five of chapter three I consider cleavability over well ordered spaces. This turns out to be a refreshing direction of study. First, unlike my previous work in cleavability, the approach did not appeal to connectivity. But also, for once cleavability moved from the premise to the conclusion of a theorem. That is, most theorems on cleavability, particularly those on cleavability over LOTS, have a certain form: If X is cleavable over Y (or a class ζ) plus additional conditions then X bears some similarity to Y (or is a member of a class not too much larger than ζ). This section furnishes an algorithm for producing nontrivial examples of cleavability, which, in the interest of parsimony, I will state here in less than full generality. If Ω ω is a minimally well ordered compactum of cardinality ℵ ω then n copies of Ω ω is cleavable over two copies of Ω ω. However if Ω 1 is the minimally well 10

11 ordered compactum of cardinality ℵ 1 then three copies of Ω 1 is not cleavable over two copies of Ω 1. In addition to the main theorems many examples are given and some open problems are posed. 11

12 Acknowledgments Earning a Ph.D in mathematics has been the most challenging and rewarding project of my life. While I feel a deep sense of pride for reaching this point, few truly achieve anything great in life alone. I have been no exception. Without my parents, Leon and Gwen Stover, I would have probably never even filled out the forms to go to college. It took me a while to decide that I wanted to dedicate my life to mathematics. But I think they knew at an early age that this was the path upon which I belonged. Until I made this determination they were patient and yet they didn t let me settle for pining away my existance in futility, but rather encouraged me to find the thing about which I could be passionate. I also thank my fiancee Mandy for her love and support these last two years. It takes a great deal of patience to listen to someone talking endlessly about something you do not understand. I am grateful to her for having this patience. I thank my dissertation advisor Dr. Arhangel skii. His status as one of the world s foremost authorities in general topology is well established. Each time we have met in the past two years, I have come a little more to appreciate the true extent of his knowledge. However he is not only a brilliant scholar, but also an excellent advisor. The direction I recieved from him was simultaneously insightful enough to fascilitate my growth as a contributor to mathematics, and yet subtle enough to provide the space I needed to personalize my work. 12

13 I also thank the members of my dissertation committee for the time they have invested in evaluating my manuscript. Finally I wish to thank myself. My hard work and dedication over the past four years has enabled me to learn more math than I ever thought I could possibly know. 13

14 Contents Abstract 3 Preface 5 Acknowledgments 12 1 π-bases π-metrizability Continuous Mappings Products Point-finite π-bases Nowhere weak-π-metrizability Coconnected Spaces and Cleavability Coconnected Spaces Cleavability over Coconnected Spaces

15 2.3 Cleavability over S Cleavability over the Class of Linearly Ordered Spaces Locally Connected Spaces Cleavability over R The Long Line Cut Points Well Ordered Spaces Bibliography 146 Appendex: Index of Terms and Notation

16 Chapter 1 π-bases A σ-discrete π-base was first observed as a necessary condition for being the absolute of a metrizable space (see [26]). H.E.White in [30] has shown that a first countable space has a dense metrizable subspace if and only if it is π-metrizable. Also Fearnley has constructed a Moore space with a σ-discrete π-base which does not densely embed into any Moore space having the Baire property [16]. It is clear that π-metrizability is preserved by open subspaces, closures of open subspaces, and dense subspaces. It is trivial to verify that every metrizable space is π-metrizable and that the Moore Plane, the Sorgenfrey Line, and βn are π- metrizable spaces which are not metrizable. Of course each of these spaces is also separable, and hence, each has a countable π-base, which is in general stronger than π-metrizability. 16

17 A space X is said to be weakly-π-metrizable if it has a σ-disjoint π-base. Weakπ-metrizability is preserved by open subspaces, closures of open subspaces, and dense subspaces in both directions. We shall however later see that every space is the closed subspace of a π-metrizable space: hence (weak)-π-metrizability is far from hereditary in general. If K is an uncountable discrete space then βk is a weakly-π-metrizable space that is not π-metrizable. A collection Γ of nonempty open subsets of X forms a uniform π base if Γ is a π-base such that for each x X, if {B : B Γ, x B} is infinite then Γ forms a base at x. A space X is uniformly π metrizable if it has a σ-discrete uniform π-base. This definition of uniformity of a π-base is a natural extension of that which is used for uniformity of a base. Recall that a space X is called d separable if there exists {K n : n N} such that each K n is a discrete (in itself) subset of X and {K n : n N} is dense in X. This property has been introduced in [3]. We introduce a stronger notion. We shall say that a space X is strongly d separable if there exists {K n : n N} such that each K n is a closed discrete subset of X and {K n : n N} is dense in X. Every π-metrizable space is strongly-d-separable. Portions of the first three sections have been submitted for publication (see [27]). 17

18 1.1 π-metrizability Most results of this section are small or technical. Some are of independent interest such as Theorem 1.18 and Theorem By pd(x) we denote the degree of psuedocompactness of X which is min{τ : τ ℵ 0 and no discrete in X family of nonempty open sets has cardinality exceeding τ}. Proposition If X is π-metrizable then π(x) pd(x). Proof Let {Ψ n : n N} be a π-base with Ψ n discrete. Each Ψ n must have size not exceeding pd(x). Thus π(x) {B n : n N} B n pd(x) ℵ 0 pd(x) = pd(x). Lemma Let A be a dense subset of X. Then X is weakly-π-metrizable if and only if A is weakly-π-metrizable. Proof Assume X is weakly-π-metrizable. Let {Ψ n : n N} be a π-base for X with Ψ n pairwise disjoint. Now consider Φ n = {B A : B Ψ n }. Then clearly Φ n is pairwise disjoint. Now let O be an open set in A. There exists O E open in X : O E A = O. So there exists U Ψn for some n such that U O E. So U A O, U A Φ n and U A as A is dense. Therefore {Φ n : n N} is a π-base for A so A is weakly-π-metrizable. 18

19 Assume A is weakly-π-metrizable. Let {Φ n : n N} be a π-base for A with Φ n pairwise disjoint. Now for each B Φ n there exists B E open in X such that B E A = B. Consider Ψn = {B E : B Φ n }. If O E,U E Ψ n and O E UE then (O E UE ) A O U, but O, U Φ n so O = U, thus O E = U E. So Ψ n is pairwise disjoint. Now let O be open in X. By regularity there exists U such that cl(u) O. There exists B Φ n for some n such that B U A. Thus B E \cl(u) is an open set disjoint from A. Thus B E cl(u) O and B E Ψ n. Therefore {Ψ n : n N)} is a π-base for X so X is weakly-π-metrizable. Corollary Let bx be a Hausdorff compactification of X where X is nowhere locally compact. Then X is weakly-π-metrizable if and only if bx\x is so. Proof By Lemma 1.12 X is weakly-π-metrizable if and only if bx is weakly-πmetrizable as X is dense in it. Now bx\x is also dense in bx thus bx is weaklyπ-metrizable if and only if bx\x is so. Theorem If Z = X Y where X and Y are weakly-π-metrizable then Z is weakly-π-metrizable. Proof Assume all closures and interiors are taken in Z. By Lemma cl(x) is weakly-π-metrizable as X is dense in it. Thus also int(cl(x)) is weakly-πmetrizable being an open subspace of a weakly-π-metrizable space. Let {Φ n : n N} be a π-base for int(cl(x)) with Φ n pairwise disjoint. Now also observe 19

20 that Y \cl(x) is an open subset of a weakly-π-metrizable space so it is weakly-πmetrizable. Let {Ψ n : n N} be a π-base for Y \cl(x) with Ψ n pairwise disjoint. We have Z = (Y \cl(x)) cl(x) with (Y \cl(x)) cl(x) =, thus Y \cl(x) is open in Z. It follows that all elements of Ψ n and Φ n are open in Z and thus in int(cl(x)) Y \cl(x). Thus ( {Φ n : n N}) ( {Ψ n : n N}) is a σ-disjoint π- base for int(cl(x)) Y \cl(x). So int(cl(x)) (Y \cl(x)) is weakly-π-metrizable. Now we need to show it is dense in Z. Let O be a nonempty open set in Z. If O cl(x) then there exists y O\cl(X) thus y Y and so y Y \cl(x). Now suppose O cl(x) then O X. Therefore Z has a dense weakly-π-metrizable subspace so Z is weakly-π-metrizable. Theorem If X = {X n : n N} has the Baire property and each X n is weakly-π-metrizable then X is weakly-π-metrizable. Proof Observe that we may write X = {cl(x n ) : n N} = ( {int(cl(x n )) : n N}) ( {cl(x n )\int(cl(x n )) : n N}). The right hand term is the countable union of closed sets with empty interior so it has empty interior. Thus the left side is dense, so it is sufficient to show it is weakly-π-metrizable. Each X n is dense in cl(x n ) so each cl(x n ) is weakly-π-metrizable. Now int(cl(x n )) is an open subspace of a weakly-π-metrizable space so it is weakly-π-metrizable. Since those sets are open in {int(cl(x n )) : n N} (as they are open in X), it follows that the elements of each π-base will remain open in {int(cl(x n )) : n N}. The union 20

21 then of these countably many σ-disjoint collections is σ-disjoint and is a π-base for {int(cl(xn )) : n N}. So X is weakly-π-metrizable. Lemma Let X be a space with a collection Γ of pairwise disjoint open sets. There exists an open dense subspace Y such that Γ Y and Γ is a discrete collection in Y. Proof Let T = {x X :any open set containing x intersects more than one element of Γ}. Then T is closed. For if t cl(t)\t there exists an open set O with t O such that O intersects at most one element of Γ. But O T so there exists y O T. So y T but O is an open set containing y that intersects only one element of Γ. This is a contradiction. Therefore T is closed and so X\T is open. Now let x Γ thus x B for some B Γ. Since Γ is a pairwise disjoint collection B is an open set containing x intersecting only one element of Γ, thus x X\T so Γ X\T. Now to show that X\T is dense. Suppose x O T where O is open. Since Γ X\T we have that O intersects no element of Γ so x / T: a contradiction. Therefore X\T is an open dense subset of X containing Γ. That Γ is a discrete collection in X\T follows from construction. Theorem Let X be a weakly-π-metrizable space with the Baire property. Then X contains a dense π-metrizable subspace. Proof Let {Ψ n : n N} be a π-base for X with Ψ n pairwise disjoint. By 21

22 Lemma there exists X n open, dense in X such that Ψ n is a discrete collection in X n. Let Y = {X n : n N}. Then Y is dense in X as X is Baire. Now define Φ n = {B Y : B Ψ n }. Certainly Φ n is discrete in Y as Ψ n is discrete in X n and Y X n. So let O be open in Y. There exists O E open in X such that O E X = O. So there exists U Ψn for some n such that U O E. So U Y Φ n and U Y O E Y = O. Since Y is dense, U Y. Thus Φn forms a π-base. It is unknown whether or not the Baire property can be dropped. It is likely that there is a weakly-π-metrizable space without a π-metrizable dense subset: the next proposition suggests where we may find one. A space X is called an L space if it is hereditarily Lindelof and inseparable. The first known L-space was constructed by Justin Moore [21]. Recently he has also constructed an L-space with a d-separable square [22]. Proposition If X is an L-space then X does not contain a dense π-metrizable subspace. Proof Such a subspace would be Lindelof and by Proposition 1.1.1, and by noting that every Lindelof space has countable degree of pseudocompactness, would be separable. Since this subspace is dense, this is a contradiction. 22

23 Theorem Every pseudocompact space with a σ-point finite π-base is weaklyπ-metrizable. Proof Let X be pseudocompact and let {Ψ n : n N} be a π-base for X with Ψ n point finite. Using regularity, for each B Ψ n choose a nonempty open set U B such that cl(u B ) B. Now let Γ n = {U B : B Ψ n }. Then {Γ n : n N} is a π-base. Well order Γ n. Construct n as follows: let B 1 = C 1. For B α, if there is no C β with β < α with C β Bα, let B α = C α. Else suppose there is a C β that intersects B α. Choose a minimal set U {C β : β < α} such that U B α. This will exist for if not then we can form a chain {U(k) : k N} {C β : β < α} and U(k) U(k + 1). Thus {cl(u(k)) : k N} as X is pseudocompact. So {cl(u(k)) : k N} is not point finite. But each U(k) = U βk for some β k < α. There is a O βk Γ n with U βk O βk. So {cl(o) : O Γ n } is not point finite, but this is a combinatorial refinement of Ψ n which is point finite: a contradiction. So we have a minimal U such that U B α. Let C α = U B α. Then let n = {C α : α I}. I claim n has the property that if A B then A B or B A. Using transfinite induction. This is clearly true for {C 1 }. Lets assume this is true for {C β : β < α}. Let A, B {C β : β α}. Suppose A B. If A, B {C β : β < α} then this is trivial as it is if A = B. So assume A = C α and B = C β where β < α. Since C α B α we have B α Cβ. So there exists a minimal element C δ for some δ < α such that C α C δ. Thus C δ Cβ and δ,β < α. Thus either 23

24 C δ C β or C β C δ. By the minimality of C δ ; C δ C β. Thus C α C β, that is A B. The result follows. Now we take surfaces of each n. Let m 1 = {maximal elements of C n } and m k = {maximal elements of C n \( {m i : i = 1,..., k 1})}. Then each m k is pairwise disjoint, for if A, B m k n and A B then A B or B A. Since both are maximal elements of the same set A = B. Furthermore for each A n there are (by point finiteness) only finitely many B n such that A B. So if there are k such elements A m k. Thus n = {m k : k N}. Thus n is σ-disjoint. Since n is a combinatorial refinement of Ψ n for each n, { n : n N} is a combinatorial refinement of { n : n N}. Thus { n : n N} is a π-base and since it is the countable union of σ-disjoint sets, it is σ-disjoint. Therefore X is weakly-π-metrizable. In [29], Uspenskiy has shown that a pseudocompact space with a σ-point finite base, is metrizable. However since a compact weakly-π-metrizable space need not be π-metrizable, our result can be improved no further. Corollary Every pseudocompact space with a σ-point finite π-base has a dense π-metrizable subspace. Proof This follows from Theorem and Theorem by noting that pseudocompact spaces have the Baire property. 24

25 Lemma Every locally finite collection of open sets in a space X has a discrete total refinement (of open sets) of the same cardinality if the collection is infinite. Proof Let X be a regular space and Ψ a locally finite collection of nonempty open sets in X. For each O Ψ there exists a nonempty open set U O O such that {V Ψ : V U O } is finite (by locally finiteness). Now let Γ = {U O : O Ψ}. Then each element of Γ intersects only finitely many elements of Γ and clearly Γ = Ψ as Γ is a combinatorial refinement. Now let n(x) = {V : x V Γ} and let N(Γ) = {n(x) : x X} (let this collection be faithful). Now N(Γ) is locally finite as for each x there is an open set O intersecting only finitely many elements of Γ and so O can only intersect those elements of N(Γ) which are intersections formed using only those sets it intersected in Γ: there are only finitely many possible. Now let m(n(γ)) = {all maximal elements (when ordered by reverse inclusion) of N(Γ)}. Note that every element in N(Γ) contains some maximal element. Let n(x) N(Γ) so n(x) = O i (i = 1,..., k), O 1 intersects only finitely many elements of Γ: say O 1,..., O n (n k) (since clearly O 1 intersects O 1,..., O k ). Then let H = {all nonempty intersections of O 1,..., O n }. Now H is a finite set so every element contains a maximal element and clearly {O i : i = 1,..., k} H. So let V H be the maximal element contained in {O i : i = 1,..., k}. Then V is maximal in N(Γ). For suppose n(y) = {U j j = 1,..., m} V O 1. Then U 1,..., U m intersect O 1, 25

26 thus U 1,..., U m {O 1,..., O n } and hence n(y) H and so V n(y). It follows then that m(n(γ)) is a total refinement of Γ. Also m(n(γ)) is locally finite as m(n(γ)) (N(Γ)). Now m(n(γ)) is pairwise disjoint. Let O, U m(n(γ)). Suppose O U. O = {O i : i = 1,..., n} and U = {U j : j = 1,..., m}. Let x O U = ( {O i : i = 1,..., n}) ( {U j : j = 1,..., m}) this means n(x) ( {O i : i = 1,..., n}) ( {U j : j = 1,..., m}) = O U. So n(x) O, thus by the maximality of O we have O = n(x) similarly n(x) = U so O = U and m(n(γ)) is pairwise disjoint. Now m(n(γ)) is a total refinement of (N(Γ)), so it is also a total refinement of Ψ and of Γ. To see that the cardinality is unchanged let g : Γ m(n(γ)) and define g(u) = V where V U(this exists as this is a total refinement). Now this is onto and by point finiteness g 1 (V ) is finite for all V m(n(γ)) thus m(n(γ)) = Γ = Ψ. Finally use regularity to choose an open set V such that cl(v O ) O for each O m(n(γ)). Then let = {V O : O m(n(γ)). Then is a discrete collection: To see this let x X. Now if x B m(n(γ)) this is trivial as m(n(γ)) is pairwise disjoint and so B is an open set containing x intersecting only V B. So otherwise by local finiteness there exists an open set O with x O such that O intersects only B 1,..., B k m(n(γ)) so O can only intersect V B1,..., V Bk. Since cl(v B ) B it follows that x O\( cl(v Bi (i = 1,..., k))) and this set intersects no element of 26

27 . Furthermore combinatorially refines m(n(γ)) (so = m(n(γ)) ) which totally refines Ψ. Thus = Ψ and totally refines Ψ. Theorem A space X is π-metrizable if and only if it has a σ-locally finite π-base. Corollary Every locally π-metrizable paracompactum is π-metrizable. Proof Let X be a locally π-metrizable paracompactum. For each x X there exists U x such that x U x with cl(u x ) O x and O x is π-metrizable. Now let Θ = {U x : x X}. By Lemma there exists a locally finite collection Γ refining Θ. Thus for each V Γ there is U x, such that V U x O x. Since O x is π-metrizable so is cl(v ) as it is the closure of an open subset. Thus cl(v ) has a σ-discrete π-base, Π, and without loss of generality we can assume Π V. Since cl(v ) is closed, Π is σ-discrete in X. So for each V Γ there exists {Ψ n (V ) : n N}, such that Ψ n (V ) is discrete in X and {Ψ n (V ) : n N} forms a π-base for V. Now define Ω n = {Ψ n (V ) : V Γ}. Then Ω n is locally finite: for if x X then there exists an open set U with x U and U intersecting only V 1,..., V k Γ. Now for each V k there exists an open set B k with x B k and B k intersecting only one element of Ψ n (V k ). Thus U B 1... Bk intersects only finitely many elements of Ω n. Thus Ω n is a σ-locally finite. That it is a π-base is straight forward. 27

28 Proposition Every π-metrizable space embeds as a dense subset in a π- metrizable Čech-complete space. Proof Let X be π-metrizable, let bx be a Hausdorff compactification of X. Now let {Ψn : n N} be a π-base for X with Ψ n discrete. As previously established, we can extend this to a σ-disjoint π-base for bx. So we have a π-base {Ψ n : n N} with Γ n pairwise disjoint and {O X : O Γ n } = Ψ n. Define T n = {x bx such that any neighborhood of x intersects more than one element of Γ n } Then T n is closed by Lemma Now to show T n is disjoint from X. Let x X, there is an open set (in X) O with x O intersecting at most one element of Ψ n. Extend O to a set O E open in bx (that is O E X = O). Then suppose OE B for some B Γn then O E B X thus O (X B) and X B Ψn. At most only one such B exists and so x / T n. Thus X bx\t n for each n N thus X {bx\t n : n N} which is π-metrizable by construction by taking as a σ-discrete π-base {U ( {bx\t n : n N}) : U {Γ n : n N)}}. Since X is dense in bx it follows that {bx\t n : n N} is also dense in bx. Thus {bx\t n : n N} is Čech-complete and certainly X is dense in it. Proposition Every pseudocompact space with a uniform π-base has a dense 1st countable subspace. 28

29 Proof Let X be a pseudocompact space and Γ a uniform π-base. Let Y = {x X : x B for infinitely many B Γ} {x X : {x} is open}. Suppose O X\Y and O is nonempty and open. By regularity (and since O contains no isolated points) there exists {O n : n N} Γ such that cl(o 1 ) O and cl(o n+1 ) O n. Now {On n N} = {cl(o n ) n N} by pseudocompactness. So there exists y {O n : n N} but y / Y so this is a contradiction. Thus Y is dense in X and is 1st countable by construction. Corollary Every pseudocompact space with a countable uniform π-base has a dense metrizable subspace. Proof Let X be a pseudocompact space with a countable uniform π-base. Then X has a dense 1st countable subspace Y. Since X has a countable π-base so does Y. Thus Y is separable. So then let D be a countable dense subset of Y. Then D is dense in X and D is first countable. Thus D is metrizable as it is countable and 1st countable. Example There exists a compact space with countable π-weight that is not first countable at any point: thus (by Proposition ) a countable π-base does not imply uniformly-π-metrizable. Proof Gleason has shown (see [17]) that there is an extremally disconnected compact space X, and an irreducible perfect onto mapping f, such that f : X [0, 1]. 29

30 Now since [0, 1] has a countable base, X has a countable π-base. Now an extremally disconnected compactum is 1st countable at a point if and only if the point is isolated. However [0, 1] has no isolated points and thus X has no isolated points. Thus X is a compact space with a countable π-base that is not first countable at any point. Proposition Every first countable separable space has a countable π-base. Example If X is a Souslin Line then X has countable Souslin number and is first countable. It is also compact, connected and perfectly normal. However X is not separable and thus is not even weakly-π-metrizable. We will later see that there is (in ZFC) a first countable space with a countable Souslin number that does not have a countable π-base. (See example in Section 1.4). Proposition Every first countable, collectionwise normal, strongly-d-separable space is weakly-π-metrizable. Proof Let X be first countable, collectionwise normal and strongly-d-separable. There exists a dense subset {D n : n N} such that each D n is discrete in X. By collectionwise normality there exists a faithfully indexed collection of pairwise disjoint open sets Ψ n such that for each x D n there exists O(x) Ψ n with x 30

31 O(x). Now suppose {B m (x) : m N} is a base at x. Define C m (x) = B m (x) O(x). Now put Φ n,m = {C m (x) : x D n }. Then Φ n,m is a pairwise disjoint collection. Now let U be a nonempty open set in X. There exists x D n for some n, such that x U, now there exists B m (x) such that B m (x) U. Thus C m (x) U. Hence {Φn,m : n, m N} is a π-base and so X is weakly-π-metrizable. Corollary Assuming the Product Measure Extension Axiom (PMEA), every first countable, normal, strongly-d-separable space is weakly-π-metrizable. Proof In [24], it was established that under PMEA, every first countable normal space is collectionwise normal. Thus the result follows from Proposition Corollary Assuming PMEA, every locally compact, perfectly normal, stronglyd-separable space is weakly-π-metrizable. Proof Every perfectly normal space is normal and every perfectly normal locally compact space is first countable so the result follows from Corollary Proposition Every Moore space with the Baire property is weakly-π-metrizable. Proof Let X be a Moore space with the Baire property. Put {Ψ n : n N} to be a development for X. That is, each Ψ n is an open cover and {st(x, Ψ n ) : n N} forms a base at x. Choose a maximal pairwise disjoint subfamily from each Ψ n, call it Φ n. Then Φ n is dense and open in X for each n. Thus { Φ n : n N} is 31

32 dense in X by the Baire property. So let O be a nonempty open set in X. There exists x Φ n for all n with x O. Now there exists n such that st(x, Ψ n ) O. So let U Φ n with x U. Then U Ψ n so U O. Thus {Φ n : n N} is a π-base and so X is weakly-π-metrizable. A space is called screenable if every open cover has a σ-disjoint open refinement. Screenability has been discussed in [9]. It is trivial to verify the following. Proposition Every screenable Moore space is weakly-π-metrizable. Recall that a space X is called subparacompact if every open cover has a σ- discrete closed refinement. Proposition Every Moore space is strongly-d-separable. Proof Every Moore space is subparacompact (see [19]). So let {Ψ n : n N} be a development for X. That is, each Ψ n is an open cover and {st(x, Ψ n ) : n N} forms a base at x. Now take a σ-discrete closed refinement Θ n for each Ψ n. Then let O be an open set with x O. There exists n such that st(x, Ψ n ) O. Now there exists F Θ n with x F. Then F U for some U Ψ n and so F O. Thus X has a σ-discrete network and thus is strongly-d-separable. Problem Does every weakly-π-metrizable space have a dense π-metrizable subspace? 32

33 1.2 Continuous Mappings This section examines the behavior of π-metrizable spaces under continuous mappings. The main results of this section are Theorem and Corollary which show that a space is the continuous image of a π-metrizable space under a perfect (closed) mapping if and only if it is strongly-d-separable. We will say a collection of sets Θ, is a π -base if every element of Θ has nonempty interior and for every nonempty open set O, there exists U Θ such that U O. Proposition Open perfect mappings preserve π-metrizability. Proof Let f : X Y be perfect onto and open and X be π-metrizable. Let {Ψn : n N} be a π-base for X with Ψ n discrete. For each set B {Ψ n : n N} there exists a closed set C B B with nonempty interior (using regularity). Then {C B : B {Ψ n : n N}} is a π -base and it is of course σ-discrete. Since f is closed and open {f(c B ) : B {Ψ n : n N}} is a π -base for Y consisting of closed sets. Let s show that this collection is σ-locally finite. Now {C B : B Ψ n } is the union of a discrete collection of closed sets so it is closed. Let y Y. The set f 1 (y) is compact so ( {C B : B Ψ n }) f 1 (y) is compact. But {B Ψ n } is an open cover of this set. So we have a finite subcover. But the cover is pairwise disjoint, so f 1 (y) must intersect only finitely many elements of Ψ n and thus of {C B : B Ψ n }. Let H = {C B : B Ψ n, f 1 (y) C B = } and let Z = H. Since H is a discrete 33

34 collection of closed sets, Z is closed and f 1 (y) Z =. Thus Y \f(z) is an open set containing y and intersecting only finitely many elements of {f(c B ) : B Ψ n } (only those not in H). Therefore {f(c B ) : B Ψ n } is locally finite and so Y has a σ-locally finite π -base. Taking interior of each element we get a σ-locally finite π-base. Then by Theorem this is equivalent to π-metrizability. Corollary If X Y is π-metrizable and Y is compact then X is π-metrizable. Proof The projection map π : X Y X is perfect and open, so this follows from Proposition Theorem A space Y is the image of a π-metrizable space X under a perfect mapping if and only if Y is strongly-d-separable. Proof Every π-metrizable space is strongly-d-separable and strong-d-separability is preserved by closed mappings: proving the first direction. Now assume Y is strongly-d-separable. Let {D n : n N} be a collection of closed discrete subspaces of Y with {D n : n N} dense in Y. Let E n = {D i : i = 1,..., n}. Then E n is closed and discrete for each n and {E n : n N} is also dense in Y. Now consider the following subspace of N Y, where N = N {p} is the Alexandroff compactification of N: X = ( {{n} E n : n N}) ({p} Y ). I claim that X is π-metrizable. Let Γ n = {{(n, d)} : d E n }. Then Γ n is discrete, for if (a, b) X with a n then (X\{n}) E n is an open set containing (a, b) and 34

35 intersecting no element of Γ n. Now if a = n then {(a, b)} is open. Furthermore {Γn : n N} is a π-base let O be a nonempty open set in X. It will be sufficient to show O intersects {{n} E n : n N}. If O ({p} Y ) = then this is trivial. So otherwise let us assume we have O = U N m for an open set U Y and N m = N \{1, 2,..., m 1}. Now since {E n : n N} is dense in Y there exists d E n for some n such that d U. Now if n m then we have {(n, d)} O where {(n, d)} Γ n. If instead n < m then E n E m and thus d E m and so {(m, d)} O where {(m, d)} Γ m. Hence {Γ n : n N} is a π-base for X and so X is π-metrizable. Now take f : X Y to be the projection map. The projection of N Y onto Y is a closed map as N is compact and X is a closed subspace of N Y thus f is a closed mapping. That f 1 (y) is compact for all y Y follows as f 1 (y) is homeomorphic to a subspace of N containing the limit point p. Thus f is a perfect mapping. In fact we need only that the mapping be closed in order that the image be strongly-d-separable and thus we get another characterization. Corollary A space Y is the image of a π-metrizable space X under a closed mapping if and only if Y is strongly-d-separable. Proposition If X is π-metrizable and f : X Y is an onto open mapping such that each fiber is compact, then Y has a σ-point-finite π-base. 35

36 Proof Let {Ψ n : n N} be a π-base for X with Ψ n discrete. Γ n = {f(b) : B Ψ n } is point-finite. For if y Y the set f 1 (y) is compact and thus every discrete collection of open sets in f 1 (y) is finite, hence it follows that f 1 (y) intersects only finitely many members of Ψ n. Thus y f(b) for only finitely many B Ψ n and so Γ n is point-finite. That {Γ n : n N} is a π-base follows trivially as f is an open mapping. Proposition If Y has an open dense π-metrizable subspace then there exist a π-metrizable space X and f : X Y such that f is onto, open and each fiber is compact. Proof Let O be the subspace. Let {Ψ n : n N} be a π-base for O with Ψ n discrete in O. Now consider subspace of N Y, where N = N {p} is the Alexandroff compactification of N: X = (N O) ({p} Y ). Now let Γ n,m = {{n} B : B Ψ m }. Then Γ n,m is discrete. For if (a, b) X and a n then X\{n} O is an open set containing (a, b) and intersecting no element of Γ n,m. If a = n then there exists and open set U O with b U and U intersecting at most one element of Ψ m. Then (a, b) {n} U and {n} U intersects at most one element of Γ n,m. Thus Γ n,m is discrete. Now let U be a basic open set in X. Then U = (V W) X where V is open in N and W is open in Y. Then there exists n N such that n V, and W O is a nonempty open subset of Y so there exists B Ψ m for some m, such that 36

37 B W O. Thus {n} B U and {n} B Γ n,m. Thus {Ψ n,m : n, m N} is a π-base for X and thus X is π-metrizable. Now consider f : X Y to be the projection mapping. Let U be a basic open set in X. Then U = (V W) X where V is open in N and W is open in Y. Then f(u) = W and so f is an open mapping. That f 1 (y) is compact for all y Y follows as f 1 (y) is homeomorphic to a subspace of N containing the limit point p. Problem How might the class of spaces described in the previous two propositions be further characterized? 1.3 Products This section seeks to determine when a product is (weakly)-π-metrizable in terms of properties of its factors. Most of the interesting results of this section follow as a corollary from Theorem These results will jointly show the extent to which (weak)-π-metrizability departs from metrizability. The first two propsitions are standard and unalarming. Proposition If X n is π-metrizable for n N, then {X n : n N} is π-metrizable. Proof Let Ψ n = {Ψ n,m : m N} be a π-base for X n with Ψ n,m discrete. There 37

38 are countably many ways to select a finite subset a 1,..., a k N. Then there are countably many ways to select n 1,..., n k N. Now let P(a 1,..., a k, n 1,..., n k ) = { {O n : n N} : O ni Ψ ai,n i for i = 1,..., k and O n = X n otherwise}. Then there are countably many such P(a 1,..., a k, n 1,..., n k ). Let x {X n : n N} for each Ψ ai,n i there exists U i open in X i such that x(i) U i and U i intersect at most one member of Ψ ai,n i. Now define U i = X i for all i 1,..., k. Then x {U n : n N} and this is an open set intersecting at most one element of P(a 1,..., a k, n 1,..., n k ). Therefore each P(a 1,..., a k, n 1,..., n k ) is discrete. Let {U n : n N} be a basic open set (U n open in X n ). Let k be such that U n = X n for all n > k. Then we can find O i,j(i) Ψ i,j(i) such that O i,j(i) U i for each i k. Then O 1,j(1)... O k,j(k) X k+1 X k+2... {U n : n N} and this is in P(1,..., k, j(1),..., j(k)). Thus this is a π-base and so {X n : n N} is π- metrizable. A similar argument would establish Corollary but it follows much smoother from established results. Corollary If X n is weakly-π-metrizable for n N, then {X n : n N} is weakly-π-metrizable. Proof Now β(x n ) is weakly-π-metrizable by Lemma Since β(x n ) has the Baire property it has a dense π-metrizable subspace Y n by Theorem Then 38

39 {Yn : n N} is π-metrizable by Proposition 1.3.1, and it is dense in {βx n : n N}. So {βx n : n N} is weakly-π-metrizable. Thus since {X n : n N} is dense in {βx n : n N} we have that {X n : n N} is weakly-π-metrizable again by Lemma Theorem If X n has a discrete collection of κ open sets for all n N and π(y ), π(x n ) κ for all n, then Y ( {X n : n N}) is π-metrizable. Proof Let X 0 = Y and let N = N {0}. Now let Ψ n be a π-base for X n for each n N with Ψ n = κ. Now let Γ n be a discrete collection open sets with Γ n = κ for all n N. We essentially want to construct almost all of the products where n factors are nontrivial: the trick is to do it for N \{n}. So we observe that there are ℵ 0 ways to choose A N \{n} such that A = n. For each k A there are κ ways to choose B k Ψ k. Thus there are κ ways to choose a set A and {B k : k A} where B k Ψ k. Now Γ n = κ, so for each A N \{n} such that A = n and {B k : k A} where B k Ψ k, we can associate a unique f(a, {B k : k A}) Γ n. So f is a one to one function. Now let O(A, {B k : k A}) = {O n : n N } where O k = B k for all k A O n = f(a, {B k : k A}) and O m = X m for all m N \(A {n}). Then O(A, {B k : k A}) is open. So let n = {O(A, {B k : k A}) : A N \{n} with A = n and B k Ψ k for each k A}. Then n is discrete. Let g {X n : n N }. Since Γ n is discrete, there exists g(n) O open in X n such that O intersects at most one elements of Γ n. Then {U n : n N } where U m = X m for n m 39

40 and U n = O, is an open set containing g. Furthermore {U n : n N } intersects B n only if O intersects π n (B) = f(a, {B k : k A}) Γ n. Since O intersects at most one element of Γ n and f is one to one, it follows that {U n : n N } intersects at most one element of n. Now to see that { n : n N} is a π-base choose a basic open set {U n : n N }, that is, U n is open for all n and U n = X n for all but finitely many n. Let B = {n N : U n X n }. Since B = n is finite, there exists m N such that n < m and m / B. Now let A N be such that A = m, m / A and B A. Now for k A choose B k Ψ k such that B k U k. Then O(A, {B k : k A}) = {On : n N } {U n : n N } as O k = B k for k A so O k U k and U k = X k for k / A so O k U k automatically. Thus { n : n N} is a σ-discrete π-base so {Xn : n N } = Y ( {X n : n N}) is π-metrizable. Corollary For every space X there exists a space Y such that X Y is π-metrizable. Proof Let D be a discrete space with D = π(x). Now let Y = D ℵ 0. Then X Y is π-metrizable by Theorem Corollary Every space is the open continuous image of a π-metrizable space. Proof Let Y be a space, and X be such that X Y is π-metrizable. Now take π : X Y Y to be the projection map. 40

41 Corollary Every space is (homeomorphic to) a closed subspace of a π-metrizable space. Proof Let Y be a space, and X be such that X Y is π-metrizable. Choose x X, then {x} Y is a closed subspace of X Y homeomorphic to X. Theorem Let {X α : α I} (N I) be a collection of not more than κ spaces with π(x α ) κ. If {λ n } is a sequence of cardinal numbers such that λ n converges to κ in the usual sense of topological ordering and X n has a discrete collection of λ n open sets for all n N, then {X α : α I} is π-metrizable. Proof From elementary set theory, there exist a partition N = {N n : n N}, such that N i Nj = if i j, and N n = ω for all n N. Then {λ n : n N i } converges to κ. We shall show that {X n : n N i } has a discrete collection of κ open sets. Write N i = {i n : n N}. Without loss of generality assume λ i1 ℵ 0. Now let Γ n be a discrete collection of open sets of X in of cardinality λ in. Choose {O n : n N} Γ i1. Now for each U Γ n for (n > 1) let h(u) = {O in : n N} where O i1 = O n, O in = U and for k 1, n put O ik = X ik. Now let n = {h(u) : U Γ n } and = { n : n N}. Then is discrete and = κ. So let Y n = {X k : k N n }. Let Y = {X α : α I\N}. It is known that π(y ) κ. Then {X α : α I} = {X α : α I\N} {X k : k N n ; n N} = Y {Y n : n N} is π-metrizable by Theorem

42 Example There exists a π-metrizable topological group that is not metrizable. Proof Let K be a discrete space with K = ℵ 1, then K is a topological group, as is K ℵ 1. Furthermore K ℵ 1 is π-metrizable from Theorem However K ℵ 1 is not metrizable. This answers a question posed in [8] Corollary Let {X α : α I} be a collection of not more that κ spaces with π(x α ) κ for all α I. If whenever {λ n } is a sequence of cardinal numbers converging to κ, there exist {X n : n N} such that X n has a collection of pairwise disjoint open sets of cardinality λ n, then {X α : α I} is weakly-π-metrizable. Proof By Lemma 1.1.6, for each X n there exists Y n dense in X n such that Y n has a discrete collection of open sets of cardinality κ. As established in Theorem 1.3.7, by taking Y α = X α for all α I\N, {Y α : α I} is π-metrizable. Since {Yα : α I} is dense in {X α : α I}, it follows that {X α : α I} is weakly-π-metrizable. Corollary For every space X there exists a compact space Y such that X Y is weakly-π-metrizable. Proof Let A be a discrete space with A = π(x). Then let B be the Alexandroff compactification of A and declare Y = B ℵ 0. Then X Y is weakly-π-metrizable by Corollary

43 Theorem Every space is the image of a weakly-π-metrizable space under an open perfect mapping. The contrast of Theorem with Section 1.2 (in particular Proposition 1.2.1) shows the distance between π-metrizable and weakly-π-metrizable. A space X has countable type if every compact subset A is contained in a compact subset C such that C has a countable base of neighborhoods. Proposition Let G be a weakly-π-metrizable space. Then bg κ \G κ is weaklyπ-metrizable for ℵ 0 κ π(g). Furthermore if G is a topological group and π(g) ℵ 0 then bg κ \G κ has a dense π-metrizable subspace for ℵ 1 κ π(g). Proof If G is compact then this is trivial. So assume G is not compact. Then G κ is nowhere locally compact. By Corollary 1.3.8, G κ is weakly-π-metrizable. Thus by Corollary 1.3.8, bg κ \G κ is weakly-π-metrizable. Now if G is a topological group then so is G κ and thus bg κ \G κ is either Lindelof or pseudocompact by [6]. Furthermore if κ > ℵ 0 then G κ is not of countable type. So from [18], bg κ \G κ is not Lindelof. Thus bg κ \G κ is pseudocompact and thus it is Baire. So by Theorem bg κ \G κ has a dense π-metrizable subspace. Theorem Let κ and λ be cardinal numbers. If Y is the product of κ factors each with at least two points and density less than or equal to λ where λ < κ, then Y is not weakly-π-metrizable. 43