Classes defined by stars and neighbourhood assignments

We apply and develop an idea of E. van Douwen used to define D-spaces. Given a topological property P, the class P^∗ dual to P (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment there is Y⊂X with Y∈P and . We prove that the classes of compact, countably compact and pseudocompact are self-dual with respect to neighbourhood assignments. It is also established that all spaces dual to hereditarily Lindelöf spaces are Lindelöf. In the second part of this paper we study some non-trivial classes of pseudocompact spaces defined in an analogous way using stars of open covers instead of neighbourhood assignments.     

On the topological Helly theorem

The main result of this paper is the following theorem, related to the missing link in the proof of the topological version of the classical result of Helly: Let be any family of simply connected compact subsets of R² such that for every i,j∈{0,1,2} the intersections Xi∩Xj are path connected and is nonempty. Then for every two points in the intersection there exists a cell-like compactum connecting these two points, in particular the intersection is a connected set.     

Compactifications of the homeomorphism group of a graph

Let Γ be a countable locally finite graph and let H(Γ) and H₊(Γ) denote the homeomorphism group of Γ with the compact-open topology and its identity component. These groups can be embedded into the space of all closed sets of Γ×Γ with the Fell topology, which is compact. Taking closure, we have natural compactifications and . In this paper, we completely determine the topological type of the pair and give a necessary and sufficient condition for this pair to be a (Q,s)-manifold. The pair is also considered for simple examples, and in particular, we find that has homotopy type of RP³. In this investigation we point out a certain inaccuracy in Sakai-Uehara's preceding results on for finite graphs Γ.     

Pseudocompact group topologies with no infinite compact subsets

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property ).Every pseudocompact Abelian group G with cardinality |G|≤2^2c satisfies this inequality and therefore admits a pseudocompact group topology with property . Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property .We also observe that pseudocompact Abelian groups with property contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.     

Characterizing continuous functions on compact spaces

We consider the following problem: given a set X and a function , does there exist a compact Hausdorff topology on X which makes T continuous? We characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant. Amongst other results, we also characterize homeomorphisms on compact metric spaces.     

Countably compact hyperspaces and Frolík sums

Let H⁰(X) (H(X)) denote the set of all (nonempty) closed subsets of X endowed with the Vietoris topology. A basic problem concerning H(X) is to characterize those X for which H(X) is countably compact. We conjecture that u-compactness of X for some u∈ω^∗ (or equivalently: all powers of X are countably compact) may be such a characterization. We give some results that point into this direction.We define the property R(κ): for every family of closed subsets of X separated by pairwise disjoint open sets and any family of natural numbers, the product is countably compact, and prove that if H(X) is countably compact for a T₂-space X then X satisfies R(κ) for all κ. A space has R(1) iff all its finite powers are countably compact, so this generalizes a theorem of J. Ginsburg: if X is T₂ and H(X) is countably compact, then so is Xn for all n<ω. We also prove that, for κ<t, if the T₃ space X satisfies a weak form of R(κ), the orbit of every point in X is dense, and X contains κ pairwise disjoint open sets, then Xκ is countably compact. This generalizes the following theorem of J. Cao, T. Nogura, and A. Tomita: if X is T₃, homogeneous, and H(X) is countably compact, then so is Xω.Then we study the Frolík sum (also called “one-point countable-compactification”) of a family . We use the Frolík sum to produce countably compact spaces with additional properties (like first countability) whose hyperspaces are not countably compact. We also prove that any product _∏α<κH⁰(Xα) embeds into .     

Topology of manifolds modeled on countable direct limits of Menger compacta

We construct n-dimensional counterparts of manifolds modeled on the space ?² equipped by the bounded weak topology (-manifolds). For -manifolds we prove the characterization, triangulation and classification theorems. In addition, a universal map of onto Q^∞ (the countable direct limit of Hilbert cubes and Z-embeddings) is constructed and characterized.     

Note on separate continuity and the Namioka property

A pair 〈B,K〉 is a Namioka pair if K is compact and for any separately continuous , there is a dense A⊆B such that f is ( jointly) continuous on A×K. We give an example of a Choquet space B and separately continuous such that the restriction fΔ_| to the diagonal does not have a dense set of continuity points. However, for K a compact fragmentable space we have: For any separately continuous and for any Baire subspace F of T×K, the set of points of continuity of is dense in F. We say that 〈B,K〉 is a weak-Namioka pair if K is compact and for any separately continuous and a closed subset F projecting irreducibly onto B, the set of points of continuity of fF_| is dense in F. We show that T is a Baire space if the pair 〈T,K〉 is a weak-Namioka pair for every compact K. Under (CH) there is an example of a space B such that 〈B,K〉 is a Namioka pair for every compact K but there is a countably compact C and a separately continuous which has no dense set of continuity points; in fact, f does not even have the Baire property.     

Variations of selective separability

A space X is selectively separable if for every sequence of dense subspaces of X one can select finite Fn⊂Dn so that is dense in X. In this paper selective separability and variations of this property are considered in two special cases: Cp spaces and dense countable subspaces in κ².     

A generalization of Gruenhage's example

We construct, assuming the continuum hypothesis, an example of nonmetrizable n-dimensional Cantor manifold Xn(n∈N) with the following properties: 1) is hereditarily separable for all k∈N; 2) is perfectly normal for every k∈N; 3) the space F(Xn) is hereditarily normal for every seminormal functor F that preserves weights and one-to-one points and such that sp(F)={1,k}; in particular, and λ₃Xn are hereditarily normal. This example is a generalization of famous Gruenhage's example given in Gruenhage and Nyikos (1993) [4].     

Symplectic duality of symmetric spaces

Let M⊂Cn be a complex n-dimensional Hermitian symmetric space endowed with the hyperbolic form ωhyp. Denote by (M^∗,ωFS) the compact dual of (M,ωhyp), where ωFS is the Fubini-Study form on M^∗. Our first result is Theorem 1.1 where, with the aid of the theory of Jordan triple systems, we construct an explicit symplectic duality, namely a diffeomorphism satisfying and for the pull-back of ΨM, where ω₀ is the restriction to M of the flat Kähler form of the Hermitian positive Jordan triple system associated to M. Amongst other properties of the map ΨM, we also show that it takes (complete) complex and totally geodesic submanifolds of M through the origin to complex linear subspaces of Cn. As a byproduct of the proof of Theorem 1.1 we get an interesting characterization (Theorem 5.3) of the Bergman form of a Hermitian symmetric space in terms of its restriction to classical complex and totally geodesic submanifolds passing through the origin.     

Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^--XF^--F⁺X+XB⁺X=0, where , and with a nonnegative matrix P, positive diagonal matrices D^±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X^∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.     

Rudin's Dowker space, strong base-normality and base-strong-zero-dimensionality

It is well known that every pair of disjoint closed subsets F₀,F₁ of a normal T₁-space X admits a star-finite open cover U of X such that, for every U∈U, either or holds. We define a T₁-space X to be strongly base-normal if there is a base B for X with |B|=w(X) satisfying that every pair of disjoint closed subsets F₀,F₁ of X admits a star-finite cover B^′ of X by members of B such that, for every B∈B^′, either or holds. We prove that there is a base-normal space which is not strongly base-normal. Moreover, we show that Rudin's Dowker space is strongly base-(collectionwise)normal. Strong zero-dimensionality on base-normal spaces are also studied.     

Spanning multi-paths in hypercubes

A set A of vertices of a hypercube is called balanced if . We prove that for every natural number n there exists a natural number π₁(n) such that for every hypercube Q with dim(Q)?π₁(n) there exists a family of pairwise vertex-disjoint paths P_i between A_i and B_i for i=1,2,…,n with if and only if {A_i,B_i∣i=1,2,…,n} is a balanced set.     

More on decompositions of edge-colored complete graphs

Let G be a family of graphs whose edges are colored with elements from a set R of r colors. We assume no two vertices of G are joined by more than one edge of color i for any i∈R, for each G∈G. will denote the complete graph with r edges joining any pair of distinct vertices, one of each of the r colors. We describe necessary and asymptotically sufficient conditions on n for the existence of a family D of subgraphs of , each of which is an isomorphic copy of some graph in G, so that each edge of appears in exactly one of the subgraphs in D.