When is c(x) a Clean Ring?

An element of a ring R is called clean if it is the sum of a unit and an idempotent and a subset A of R is called clean if every element of A is clean. A topological characterization of clean elements of C(X) is given and it is shown that C(X) is clean if and only if X is strongly zero-dimensional, if and only if there exists a clean prime ideal in C(X). We will also characterize topological spaces X for which the ideal C_K(X) is clean. Whenever X is locally compact, it is shown that C_K(X) is clean if and only if X is zero-dimensional.     

Calibers and point-finite cellularity of the space Cp(X) and some questions of S. Gul'ko and M. Husek

It is well known that ℵ₁ is a precaliber of Cp(X) for every Tychonoff space X. We prove under GCH that a compact space X is metrizable if and only if ℵ₁ and ℵ₂ are calibers of Cp(X). We show also that if X is a compact space then ℵ₁ is a caliber of Cp(X) if and only if its diagonal Δ ⊂ X² is small in the sense of Husek [9]. A similar method is used to establish that if X is an extremally disconnected compact space then Cp(X) admits a continuous injection into Σ₁ (τ) (for some τ) if and only if the space X is separable.     

Covering properties on X2⧹Δ, W-sets,and compact subsets of Σ-products

Let Δ ? X¹ be the diagonal. In the first part of this paper, we show that a compact space X is Corson compact (resp., Eberlein compact; compact metric) if and only if X²?Δ is metalindelöf (resp., σ-metacompact; paracompact). In the second part of the paper, we investigate the notion of a W-set in a space X, which is defined in terms of an infinite game. We show that a compact space X is Corson compact if and only if X has a W-set diagonal, and that a compact scattered space X is strong Eberlein compact if and only if each point of X is a W-set in X.     

Ordinals and set-valued zero-selections for hyperspaces

A continuous zero-selection f for the Vietoris hyperspace F(X) of the nonempty closed subsets of a space X is a Vietoris continuous map f:F(X)→X which assigns to every nonempty closed subset an isolated point of it. It is well known that a compact space X has a continuous zero-selection if and only if it is an ordinal space, or, equivalently, if X can be mapped onto an ordinal space by a continuous one-to-one surjection. In this paper, we prove that a compact space X has an upper semi-continuous set-valued zero-selection for its Vietoris hyperspace F(X) if and only if X can be mapped onto an ordinal space by a continuous finite-to-one surjection.     

Even continuity and topological equicontinuity in topologized semigroups

A topologized semigroup X having an evenly continuous resp., topologically equicontinuous, family RX of right translations is investigated. It is shown that: (1) every left semitopological semigroup X with an evenly continuous family RX is a topological semigroup, (2) a semitopological group X is a paratopological group if and only if the family RX is evenly continuous and (3) a semitopological group X is a topological group if and only if the family RX is topologically equicontinuous. In particular, we get that for any paratopological group X which is not a topological group, the family RX provides an example of a transitive group of homeomorphisms of X that is evenly continuous and not topologically equicontinuous. The last conclusion answers negatively a question posed by H.L. Royden.     

Almost rimcompact spaces

A 0-space is a completely regular Hausdorff space possesing a compactification with zero-dimensional remainder. Recall that a space X is called rimcompact if X has a basis of open sets with compact boundaries. It is well known that X is rimcompact if and only if X has a compactification which has a basis of open sets whose boundaries are contained in X. Thus any rimcompact space is a 0-space; the converse is not true. In this paper the class of almost rimcompact spaces is introduced and shown to be intermediate between the classes of rimcompact spaces and 0-spaces. It is shown that a space X is almost rimcompact if and only if X has a compactification in which each point of the remainder has a basis (in the compactification) of open sets whose boundaries are contained in X.     

Domain representability of certain function spaces

Let Cp(X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp(X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.     

Construction of quotient BCI(BCK)-algebra via a fuzzy ideal

The present paper gives a new construction of a quotient BCI(BCK)-algebraX/μ by a fuzzy ideal μ inX and establishes the Fuzzy Homomorphism Fundamental Theorem. We show that if μ is a fuzzy ideal (closed fuzzy ideal) ofX, thenX/μ is a commutative (resp. positive implicative, implicative) BCK (BCI)-algebra if and only if μ is a fuzzy commutative (resp positive implicative, implicative) ideal ofX. Moreover we prove that a fuzzy ideal of a BCI-algebra is closed if and only if it is a fuzzy subalgebra ofX. We show that if the period of every element in a BCI-algebraX is finite, then any fuzzy ideal ofX is closed. Especially, in a well (resp, finite, associative, quasi-associative, simple) BCI-algebra, any fuzzy ideal must be closed.     

A characteristic property of the exponential distribution

The following characterization of the exponential distribution is given: Under suitable conditions on the random variables X and Y, X is exponentially distributed if and only if E[min{X, Y}]=E(X)P(X<Y).     

Basic sequences and norming subspaces in non-quasi-reflexive Banach spaces

A Banach spaceX is non-quasi-reflexive (i.e. dimX ^**/X=∞) if and only if it contains a basic sequence spanning a non-quasi-reflexive subspace. In fact, this basic sequence can be chosen to be non-k-boundedly complete for allk. A basic sequence which is non-k-shrinking for allk exists inX if and only ifX ^* contains a norming subspace of infinite codimension. This need not occur even ifX is non-quasi-reflexive. Every norming subspace ofX ^* has finite codimension if and only if for every normingM inX ^*, everyM-closedY inX,M∩Y ^T is norming overX/Y. This solves a problem due to Schäffer [19].     

A generalization of Hiraguchi's: Inequality for posets

For a poset X, Dim(X) is the smallest positive integer t for which X is isomorphic to a subposet of the cartesian product of t chains. Hiraguchi proved that if | X | ? 4, then Dim(X) ? [| X |/2]. For each k ? 2, we define Dim_k(X) as the smallest positive integer t for which X is isomorphic to a subposet of the cartesian product of t chains, each of length k. We then prove that if | X | ? 5, Dim₃(X) ? {| X |/2} and if | X | ? 6, then Dim₄(X) ? [| X |/2].     

Submaximal and door compactifications

In this paper, a characterization is given for compact door spaces. We, also, deal with spaces X such that a compactification K(X) of X is submaximal or door.Let X be a topological space and K(X) be a compactification of X.We prove, here, that K(X) is submaximal if and only if for each dense subset D of X, the following properties hold:

(i)

D is co-finite in K(X);

(ii)

for each x∈K(X)?D, {x} is closed.

If X is a noncompact space, then we show that K(X) is a door space if and only if X is a discrete space and K(X) is the one-point compactification of X.     

A note in approximative compactness and midpoint locally k-uniform rotundity in Banach spaces

In this article, we prove the following results: (1) A Banach space X is weak midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively weakly compact k-Chebyshev set; (2) A Banach space X is midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively compact k-Chebyshev set.     

Algebras of real-valued continuous functions in product spaces

Let {X_i:iϵI} be an arbitrary family of spaces, we say that the cartesian product X has the approximation property when C(X) coincides with the Algebra on X generated by the functions which depend on one variable. In this paper we study the problem of characterizing topologically when an arbitrary product space has the approximation property. We prove that if X is an uncountable pseudo-ℵ₁-compact P-space, then X×Y has the approximation property if, and only if, X×Y is pseudo-ℵ₁-compact. As a corollary we obtain the following characterization for P-spaces: Let X and Y be P-spaces, then X×Y has the approximation property if, and only if, X or Y is countable or X×Y is pseudo-ℵ₁-compact.     

Local selections and local dendrites

Let X be a Peano continuum, C(X) its space of subcontinua, and C(X, ε) the space of subcontinua of diameter less than ε. A selection on some subspace of C(X) is a continuous choice function; the selection σ is rigid if σ(A) ? B ? A implies σ(A) = σ(B). It is shown that X is a local dendrite (contains at most one simple closed curve) if and only if there exists ε > 0 such that C(X, ε) admits a selection (rigid selection). Further, C(X) admits a local selection at the subcontinuum A if and only if A has a neighborhood (relative to the space C(X)) which contains no cyclic local dendrite; moreover, that local selection may be chosen to be a constant.     

The tangential end fibration of an aspherical Poincaré complex

We construct a sphere fibration over a finite aspherical Poincaré complex X, which we call the tangential end fibration, under the condition that the universal cover of X is forward tame and simply connected at infinity. We show that it is tangent to X if the formal dimension of X is even or, when the formal dimension is odd, if the diagonal X→X×X admits a Poincaré embedding structure.     

On nearly pseudocompact spaces

A completely regular space X is called nearly pseudocompact if υX?X is dense in βX?X, where βX is the Stone-?ech compactification of X and υX is its Hewitt realcompactification. After characterizing nearly pseudocompact spaces in a variety of ways, we show that X is nearly pseudocompact if it has a dense locally compact pseudocompact subspace, or if no point of X has a closed realcompact neighborhood. Moreover, every nearly pseudocompact space X is the union of two regular closed subsets X₁, X₂ such that Int X₁ is locally compact, no points of X₂ has a closed realcompact neighborhood, and $\text{Int(X}{}_1\text{?X}{}_2\text{)=?}$. It follows that a product of two nearly pseudocompact spaces, one of which is locally compact, is also nearly pseudocompact.     

On projections and limit mappings of inverse systems of compact spaces

A compactificaton αX of a completely regular space X is “determined” by a subset F of C¹(X) if αX is the smallest compactificaton of X to which each element of F extends, and is “generated” by F if the evaluation map $\text{e}{}_{\mathrm{F}}\text{:X →}\text{R}{}^{\mathrm{n}}\text{,n = |F|}$, is an embedding and $\text{αX =}\text{e}{}_{\mathrm{F}}\text{(X)}$. Evidently, if F either determines or generates αX, then every elements of F has an extension to αX; whenever F satisfies this latter condition, the set of all such extensions is denoted F^α.A major results of our previous paper is that F determines αX if and only if F^α separates points of αX ? X. A major result of the present paper is that F generates αX if and only if F^α separates points of αX.     

A pair of spaces of upper semi-continuous maps and continuous maps

For a Tychonoff space X, we use ↓USC(X) and ↓C(X) to denote the families of the regions below all upper semi-continuous maps and of the regions below all continuous maps from X to I=[0,1], respectively. In this paper, we consider the spaces ↓USC(X) and ↓C(X) topologized as subspaces of the hyperspace Cld(X×I) consisting of all non-empty closed sets in X×I endowed with the Vietoris topology. We shall prove that ↓USC(X) is homeomorphic (≈) to the Hilbert cube Q=ω[−1,1] if and only if X is an infinite compact metric space. And we shall prove that (↓USC(X),↓C(X))≈(Q,c₀), where , if and only if ↓C(X)≈c₀ if and only if X is a compact metric space and the set of isolated points is not dense in X.     

Super edge-connectivity of mixed Cayley graph

A graph X is max-λ if λ(X)=δ(X). A graph X is super-λ if X is max-λ and every minimum edge-cut set of X isolates one vertex. In this paper, we proved that for all but a few exceptions, the mixed Cayley graph which is defined as a new kind of semi-regular graph is max-λ and super-λ.