Hausdorff compactifications and lebesgue sets

Let X be a completely regular Hausdorff space and let H be a subset of C¹(X) which separates points and closed sets. By embedding X into a cube whose factors are indexed by H, a Hausdorff compactification e_HX of X is obtained. Given two subsets F and G of C¹(X) which separate points from closed sets, in the present paper we obtain a necessary and sufficient condition for the equivalence of e_FX and e_GX. The condition is expressed in terms of the space X and the sets F and G alone, herewith solving a question raised by Chandler.     

Building suitable sets for locally compact groups by means of continuous selections

If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [K.-H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1-2) (1990) 181-194] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory.     

An intersection theorem for four sets

Fix integers n,r?4 and let F denote a family of r-sets of an n-element set. Suppose that for every four distinct A,B,C,D∈F with |A∪B∪C∪D|?2r, we have A∩B∩C∩D≠∅. We prove that for n sufficiently large, , with equality only if _?F∈FF≠∅. This is closely related to a problem of Katona and a result of Frankl and Füredi [P. Frankl, Z. Füredi, A new generalization of the Erd?s-Ko-Rado theorem, Combinatorica 3 (3-4) (1983) 341-349], who proved a similar statement for three sets. It has been conjectured by the author [D. Mubayi, Erd?s-Ko-Rado for three sets, J. Combin. Theory Ser. A, 113 (3) (2006) 547-550] that the same result holds for d sets (instead of just four), where d?r, and for all n?dr/(d−1). This exact result is obtained by first proving a stability result, namely that if |F| is close to then F is close to satisfying _?F∈FF≠∅. The stability theorem is analogous to, and motivated by the fundamental result of Erd?s and Simonovits for graphs.     

Some upper bounds for density of function spaces

Let Cα(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X which is closed under finite unions. We proved that the density of the space Cα(X,Y) is at most iw(X)⋅d(Y) where iw(X) denotes the i-weight of the Tychonoff space X, and d(Y) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function Ψ, and Y has a base consists of Ψ-convex subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal κ, there is a pathwise connected space Y such that π-weight of Y is κ, but Souslin number of the space Ck([0,1],Y) is κ².     

The D-property of some Lindelöf spaces and related conclusions

It is shown that the space Cp(τω) is a D-space for any ordinal number τ, where . This conclusion gives a positive answer to R.Z. Buzyakova's question. We also prove that another special example of Lindelöf space is a D-space. We discuss the D-property of spaces with point-countable weak bases. We prove that if a space X has a point-countable weak base, then X is a D-space. By this conclusion and one of T. Hoshina's conclusion, we have that if X is a countably compact space with a point-countable weak base, then X is a compact metrizable space. In the last part, we show that if a space X is a finite union of θ-refinable spaces, then X is a αD-space.     

On function spaces topologies in the setting of ?ech closure spaces

We consider different types of topologies on the set of functions between two ?ech closure spaces and investigate some of their properties.     

The smallest basically disconnected preimage of a space

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

Compactifications determined by subsets of C1(X)

It is well known that every compactification of a completely regular space X can be generated, via a Tychonoff-type embedding, by some suitably chosen subset of C¹(X). Different subsets may give rise to equivalent compactifications, and we are concerned with the problem of finding all subsets of C¹(X) which yield a given compactification αX. The problem is easier if generalized: we say that a subset F of C¹(X) “determines” the compactification αX if αX is the smallest compactification to which every element of F extends, and give a simple necessary and sufficient condition for F to determine a given compactification αX. A number of sufficient conditions for two sets to determine the same compactification are given, and the relation between sets which determine αX and those which generate αX (via an embedding) is considered. Generally, a much smaller set of functions is required to determine αX than to generate it; the number needed to determine αX is never more than the weight of αX?X, while the number required to generate it is, if infinite, equal to the weight of αX.     

Zero-dimensional proximities and zero-dimensional compactifications

We introduce zero-dimensional proximities and show that the poset 〈Z(X),?〉 of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space X is isomorphic to the poset 〈Π(X),?〉 of zero-dimensional proximities on X that induce the topology on X. This solves a problem posed by Leo Esakia. We also show that 〈Π(X),?〉 is isomorphic to the poset 〈B(X),⊆〉 of Boolean bases of X, and derive Dwinger's theorem that 〈Z(X),?〉 is isomorphic to 〈B(X),⊆〉 as a corollary. As another corollary, we obtain that for a regular extremally disconnected space X, the Stone-?ech compactification of X is a unique up to equivalence extremally disconnected compactification of X.     

Counting substructures I: Color critical graphs

Let F be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of F in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits (1968) [8], who proved that there is one copy of F, and of Rademacher, Erd?s (1962) [1] and [2] and Lovász and Simonovits (1983) [4], who proved similar counting results when F is a complete graph.One of the simplest cases of our theorem is the following new result. There is an absolute positive constant c such that if n is sufficiently large and 1?q<cn, then every n vertex graph with ⌊n²/4⌋+q edges contains at least

     

Gated sets in metric spaces

The concept of a gated subset in a metric space is studied, and it is shown that properties of disjoint pairs of gated subsets can be used to investigate projections in Tits buildings.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

The construction of all locally compact extensions

The paper presents one of the ways to construct all the locally compact extensions of a given Tychonoff space T. First, there proved the “local” variant of the Stone-C?ech theorem on “completely regular” Riesz spaces X(T) of continuous bounded functions on T with no unit function, in general, but with a collection of local units. In Theorem 1 it is proved that all the functions from X(T) can be “completely regularly” extended on the largest locally compact extension β_xT. Theorem 3 states, that β_xT are presenting, in fact, all the locally compact extensions of T.     

On tree‐decompositions of one‐ended graphs

A graph is one‐ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex v dominates a ray in the end if there are infinitely many paths connecting v to the ray such that any two of these paths have only the vertex v in common. We prove that if a one‐ended graph contains no ray which is dominated by a vertex and no infinite family of pairwise disjoint rays, then it has a tree‐decomposition such that the decomposition tree is one‐ended and the tree‐decomposition is invariant under the group of automorphisms. This can be applied to prove a conjecture of Halin from 2000 that the automorphism group of such a graph cannot be countably infinite and solves a recent problem of Boutin and Imrich. Furthermore, it implies that every transitive one‐ended graph contains an infinite family of pairwise disjoint rays.     

Property C. Wallman basis and S-metrizability

In this present paper we prove that every Lindelof space which has a perfect locally connected Hausdorff compactification, has property C. (This latter concept was introduced by R.F. Dickman Jr). We make clear that this class of Lindelöf spaces properly contains the class of paracompact, connected, locally compact and locally connected spaces, as well as the class of those spaces whose topology can be induced by a metric with property S (or S-metrizable spaces). In this fashion, we simultaneously generalize two previous results of Dickman on spaces with property C. The use of Wallman basis with certain connectedness properties turns out to be a very convenient tool in the construction of locally connected compactifications as well as in characterizing S-metrizable spaces.     

Bezoutians, Euclidean Algorithm, and Orthogonal Polynomials

We prove a quadratic expression for the Bezoutian of two univariate polynomials in terms of the remainders for the Euclidean algorithm. In case of two polynomials of the same degree, or of consecutive degrees, this allows us to interpret their Bezoutian as the Christoffel- Darboux kernel for a finite family of orthogonal polynomials, arising from the Euclidean algorithm. We give orthogonality properties of remainders, and reproducing properties of Bezoutians. Received December 13, 2004     

Function spaces over GO-spaces: Part I

We describe the structure of spaces of continuous step functions over GO-spaces. We establish a relation between the Dedekind completion of a GO-space L and properties of the space of continuous functions from L to 2 with finitely many steps. We use the established relation to prove that a countably compact GO-space L has Lindelöf Cp(L) iff the Dedekind remainder of L is Lindelöf and every compact subspace of L is metrizable. Or equivalently, a countably compact GO-space L has Lindelöf Cp(L) iff every compact subspace of L is metrizable and a Gδ-set in L. Other results are obtained.     

Local connectedness and extension of uniformly continuous functions

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. The straight spaces have been studied in [A. Berarducci, D. Dikranjan, J. Pelant, An additivity theorem for uniformly continuous functions, Topology and its Applications 146-147 (2005) 339-352], which contains characterization of the straight spaces within the class of the locally connected spaces (they are the uniformly locally connected ones) and the class of the totally disconnected spaces (they coincide with the totally disconnected Atsuji spaces). We show that the completion of a straight space is straight and we characterize the dense straight subspaces of a straight space. In order to clarify further the relation between straightness and the level of local connectedness of the space we introduce two more intermediate properties between straightness and uniform local connectedness and we give various examples to distinguish them. One of these properties coincides with straightness for complete spaces and provides in this way a useful characterization of complete straight spaces in terms of the behaviour of the quasi-components of the space.     

Some results on separate and joint continuity

Let f:X×K→R be a separately continuous function and C a countable collection of subsets of K. Following a result of Calbrix and Troallic, there is a residual set of points x∈X such that f is jointly continuous at each point of {x}×Q, where Q is the set of y∈K for which the collection C includes a basis of neighborhoods in K. The particular case when the factor K is second countable was recently extended by Moors and Kenderov to any ?ech-complete Lindelöf space K and Lindelöf α-favorable X, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when K is a Lindelöf p-space and X is conditionally σ-α-favorable space. Here we add new results of this sort when the factor X is σC(X)-β-defavorable and when the assumption “base of neighborhoods” in Calbrix-Troallic's result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces.     

Graphs with prescribed degree sets and girth

For a finite nonempty set of integers, each of which is at least two, and an integern 3, the numberf(;n) is defined as the least order of a graph having degree set and girthn. The numberf(;n) is evaluated for several sets and integersn. In particular, it is shown thatf({3, 4}; 5) = 13 andf({3, 4}; 6) = 18.Research of the third author was partially supported by a Faculty Research Fellowship from Western Michigan University.