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For every space $X$ let $\mathcal K(X)$ be the set of all compact subsets of $X$. Christensen \cite{c:74} proved that if $X, Y$ are separable metrizable spaces and $F\colon\mathcal{K}(X)\to\mathcal{K}(Y)$ is a monotone map such that any $L\in\mathcal{K}(Y)$ is covered by $F(K)$ for some $K\in\mathcal{K}(X)$, then $Y$ is complete provided $X$ is complete. It is well known \cite{bgp} that this result is not true for non-separable spaces. In this paper we discuss some additional properties of $F$ which guarantee the validity of Christensen's result for more general spaces., Comment: 11 pages

For every triple of integers a, b, and c, such that a>O, b>0, and c>1, there is a homogeneous, non-bihomogeneous continuum whose every point has a neighborhood homeomorphic the Cartesian product of three Menger compacta m^a, m^b, and m^c. In particular, there is a homogeneous, non-bihomogeneous, Peano continuum of covering dimension four., 9 pages

We solve two problems posed by E. Michael (1981) concerning whether certain compact-covering maps are inductively perfect, we generalize a theorem of A. Ostrovsky on a condition under which tri-quotient maps are inductively perfect and we present an example which answers a more recent question of Michael in the same general area.

The first three parts of a theorem of Effros have been crucial to recent investigations in continua theory. The purpose of this paper is to examine some other parts of this theorem and apply them to continua theory as well. Here is one such application. A Borel transversal to the composants of an indecomposable continuum X is a Borel subset B of X such that B intersects each composant of X in exactly one point. It has been unknown for any indecomposable continuum whether or not it has a Borel transversal to its composants. In this paper it is shown that solenoids and Knaster continua do not have Borel transversals to their composants.

Given a Tychonoff space X and classes U and V of topological groups, we say that a topological group G = G ( X , U , V ) is a free ( U , V )-group over X if (a) X is a subspace of G , (b) G ϵ U , and (c) every continuous f : X → H with H ϵ V extends uniquely to a continuous homomorphism f: G → H . For certain classes U and V , we consider the question of the existence of free ( U , V )- groups. Our principal results are the following. Let PA and CA denote, respectively, the class ofpseudocompact Abelian groups and the class of compact Abelian groups. Then 1. (a) there is a free ( PA , PA )-group over X iff; X = O and 2. (b) there is for each X a free ( PA , CA )-group over X in which X is closed.

We construct a quasi-metric space that does not admit any complete quasi-uniformity.

We solve or partially solve a number of problems of Watson concerning the preservation of normality by forcing. We also provide general machinery for applying reflection methods in the class of spaces of pointwise countable type.

We investigate the properties monolithic and d-separable for the hyperspace H(X) of all nonempty closed subsets of a compact Hausdorff space X . A. Arhangelskii has asked whether H(X) monolithic is equivalent to X metrizable. We answer this with: Let X be a compact orderable space. Then H(X) is monolithic iff X is monolithic and hereditarily Lindelof. So, a Suslin continuum has a monolithic hyperspace. In contrast, MA(ω 1 ) implies that for any compact Hausdorff space X , H(X) is monolithic iff X is metrizable. We prove that H(X) is always d-separable. A special case of this yields that every locally compact Hausdorff space X has a discrete (in H(X) ) π-net.

Metrizability is an extremely strong property where trees are concerned, and it turns out that in many ways, monotone normality is the appropriate generalization when the trees have uncountable chains. We show that monotone normality is equivalent to the tree being the topological direct sum of ordinal spaces, each of which is a convex chain in the tree. Several metrization theorems are proven, some in ZFC, some just assuming ZF or “ZF + Countable AC”, and still others assuming ZFC-independent axioms, as well as theorems in a similar spirit with monotone normality of the tree as a conclusion. The property of being collectionwise Hausdorff plays a key role, and we obtain partial results on the still unsolved problems of whether it is consistent that every collectionwise Hausdorff tree or every normal tree is monotone normal.

We prove a Ramsey-style theorem for sequences of vectors in an infinite-dimensional vector space over a finite field. As an application of this theorem, we prove that there are countably infinite Abelian groups whose Bohr topologies are not homeomorphic.

We prove that every homeomorphism of the pseudo-arc is conjugate to an e-homeomorphism.

Nielsen coincidence theory is extended to manifolds with boundary. For X and Y compact connected oriented n -manifolds with boundary, and for maps ƒ : X → Y and g : ( X , ∂ X ) → ( Y , ∂ Y ), a coincidence index—which is a local version of Nakaoka's Lefschetz coincidence number—and a Nielsen coincidence number are defined and their properties explored. As an application, coincidence-producing maps g are characterized if Y is acyclic over the rationals and many new examples of coincidence-producing maps are constructed.

K.R. Kellum has proved that a continuum is an almost continuous image of the interval [0, 1] if and only if it is an almost Peano continuum. Hence, a continuum is an almost continuous image of [0, 1] if it has a dense arc component. Our principal result is that any almost arcwise connected, semi-hereditarily unicoherent, metric continuum with only countably many arc components has a dense arc component. An example is given to show that this is not true for unicoherent continua in general. It is also shown that any semi-hereditarily unicoherent continuum with only countably many arc components has at most one dense arc component, and if it has a dense arc component, then every other arc component is nowhere dense. This generalizes results of Fugate and Mohler for λ-dendroids.

A space X is said to be consonant if, on the set of closed subsets of X, the upper Kuratowski topology coincides with the co-compact topology. It is known that Cech-complete spaces are consonant and that consonance is neither preserved by Gδ, subsets nor stable under products. We show that all Gδ subspaces of a consonant space X are consonant if the Vietoris topology on compact subsets of X is hereditarily Baire; and that is always the case if all compact subspaces of X are separable and of countable character in X. Spaces which are Gδ subspaces of consonant paracompact p-spaces are also shown to be consonant. Concerning products, we show that the product of a consonant paracompact p-space and a Cech-complete space is consonant. We also answer some questions of Nogura and Shakhmatov related to product and topological sum operations in the class of regular consonant spaces.

We continue to investigate various diagonalization properties for sequences of open covers of separable metrizable spaces introduced in Part I. These properties generalize classical ones of Rothberger, Menger, Hurewicz, and Gerlits-Nagy. In particular, we show that most of the properties introduced in Part I are indeed distinct. We characterize two of the new properties by showing that they are equivalent to saying all finite powers have one of the classical properties above (Rothberger property in one case and in Menger property in the other). We consider for each property the smallest cardinality of a metric space which fails to have that property. In each case this cardinal turns out to equal another well-known cardinal less than the continuum. We also disprove (in ZFC) a conjecture of Hurewicz which is analogous to the Borel conjecture. Finally, we answer several questions from Part I concerning partition properties of covers.

We consider the intersection topology formed by the ordinal topology on ω 1 and a separable metric topology. Such a space is never perfectly normal, but it can be normal or perfect; which one depends on the model of set theory and the choice of the metric topology.

It is proven that every homeomorphism of a product of pseudo-arcs is a composition of a product homeomorphism with a homeomorphism which only permutes the factors

The purpose of this paper is to determine which metric spaces X and Y are such that the uniformly continuous maps f : X → Y are precisely the continuous maps between ( X , τ 1 ) and ( Y , τ 2 ) for some new topologies τ 1 and τ 2 on X and Y respectively.

International audience; A topology T on a set X is called consonant if the Scott topology of the lattice T is compactly generated; equivalently, if the upper Kuratowski topology and the co-compact topology on closed sets of X coincide. It is proved that every completely regular consonant space is a Prohorov space, and that every first countable regular consonant space is hereditarily Baire. If X is metrizable separable and co-analytic, then X is consonant if and only if X is Polish. Finally, we prove that every pseudocompact topological group which is consonant is compact. Several problems of Dolecki, Greco and Lechicki, of Nogura and Shakmatov, are solved.

In 1971 H. Tamano asked the question: Is a space X paracompact if X has a closure-preserving cover by compact closed sets? From this came many results concerning spaces with various types of closure-preserving covers as well as new questions about spaces having these properties. In this paper we generalize many known results by considering spaces with closure-preserving J-covers, where J is any ideal of closed subsets. Several characterization theorems are also obtained linking the properties of J-scattered spaces, hereditarily metacompact spaces, spaces with special closure-preserving J-covers, and spaces defined by certain topological games.

We show that the Lindelof number of the product of ℵ copies of the one-point Lindelofization of any discrete space is ℵ. We give a combinatorial proof of this result and a careful proof by elementary submodels which provides an introduction for general topologists to the use of elementary submodels.

A generalization to higher cardinals of a variant of Martin's axiom is considered. Numerous applications are given in set theory and in set-theoretic topology.