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1.
《Quaestiones Mathematicae》2013,36(8):1091-1099
Abstract

Given a space X, we will say that a class of subsets of X is dominated by a class ? if for any A, there exists a B? such that A ? . In particular, all (closed) discrete subsets of X are countably dominated (which we frequently abbreviate as ω-dominated) if, for any (closed) discrete set D ? X, there exists a countable set B ? X such that D ? . In this paper, we investigate the topological properties of spaces in which (closed) discrete subspaces are dominated either by countable subsets or by Lindelöf subspaces.  相似文献   

2.
We present a study of two versions of the point-picking game defined by Berner and Juhasz. Given a space X there are two rivals O and P who take turns playing on X. In the n-th round Player O takes a non-empty open subset Un of the space X and P responds by choosing a point xnUn. After ω-many moves are completed, the family is called the play of the game. In the CD-game CD(X) Player P wins if the set is closed and discrete. Otherwise O is the winner. In the CL-game CL(X, p), where the point pX is fixed, Player O wins if contains p in its closure. If , then P is declared to be the winner. We show that in spaces Cp(X) both CD-game and CL-game are equivalent to Gruenhage’s W-game for Player O. If , then Player O has a winning strategy in CL(X, p). The converse is not always true. However, if X is separable or compact of π-weight ≤ ω1, then existence of a winning strategy for O in CL(X, p) is equivalent to .  相似文献   

3.
《Quaestiones Mathematicae》2013,36(5):593-604
Abstract

Let X be a topological space and Cc(X) be the functionally countable subalgbera of C(X). We call X to be a countably uniform closed-space, briefly, a CU C-space, if Cc(X) is closed under uniform convergence. We investigate that countably uniform closedness need not closed under finite intersection and infinite product. It is shown that if X is a countable union of quasi-components, then X is a CU C-space. We characterize Cc-embedding and also -embedding in CU C-spaces. A subset S of X is called Zc-embedded, if each ZZc(S) is the restriction of a zero-set of Zc(X). It is observed that in a zero-dimensional CU C-space, each Lindelöf subspae is Zc-embedded. Moreover, it is shown that in CU C-spaces, each Lindelöf subspace is Cc-embedded if and only if it is c-completely separated from each zero-set, which is disjoint from it. Also in latter spaces, it is observed that for each S ? X, Cc-embedding, -embedding and Zc-embedding coincide, when S belongs to Zc(X) or it is a c-pseudocompact space. Finally, when X is both a CU C-space and a CP-space, then each Zc-embedded subspace is Cc-embedded (-embedded) in X.  相似文献   

4.
《Quaestiones Mathematicae》2013,36(3-4):273-290
Abstract

A bispace is called strongly zero-dimensional if its bispace Stone—?ech compactification is zero—dimensional. To motivate the study of such bispaces we show that among those functorial quasi—uniformities which are admissible on all completely regular bispaces, some are and others are not transitive on the strongly zero-dimensional bispaces. This is in contrast with our result that every functorial admissible uniformity on the completely regular spaces is transitive precisely on the strongly zero-dimensional spaces.

We then extend the notion of strong zero-dimensionality to frames and biframes, and introduce a De Morgan property for biframes. The Stone—Cech compactification of a De Morgan biframe is again De Morgan. In consequence, the congruence biframe of any frame and the Skula biframe of any topological space are De Morgan and hence strongly zero-dimensional. Examples show that the latter two classes of biframes differ essentially.  相似文献   

5.
6.
In this paper, we show that assuming ω1 < , there exists a Tychono? star-C-Menger space having a regular-closed Gδ-subspace which is not star-C-Menger which gives a partial answer to a question of Song and Yin [12], and continue to investigate topological properties of star-C-Menger spaces.  相似文献   

7.
《Quaestiones Mathematicae》2013,36(8):1079-1090
Abstract

For any ideal of closed sets in X, let be the family of those functions in C(X) whose support lie on . Further let contain precisely those functions f in C(X) for which for each ? > 0, {xX: |f (x)| ≥ ?} is a member of . Let stand for the set of all those points p in βX at which the stone extension f? for each f in is real valued. We show that each realcompact space lying between X and βX is of the form if and only if X is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally- or almost locally-, becomes a space of the same form. We further show that is a free ideal (essential ideal) of C(X) if and only if is a free ideal (essential ideal) of when and only when X is locally- (almost locally-). We address the problem, when does or become identical to the socle of the ring C(X). The results obtained turn out to imply a special version of the fact obtained by Azarpanah corresponding to the choice the ideal of compact sets in X. Finally we observe that the ideals of the form of C(X) are no other than the z?-ideals of C(X).  相似文献   

8.
《Quaestiones Mathematicae》2013,36(6):765-779
Abstract

Assume that is an ideal on ?, and ∑n xn is a divergent series in a Banach space X. We study the Baire category, and the measure of the set A() := {t ∈ {0, 1}?: ∑n t(n)xn is -convergent}. In the category case, we assume that has the Baire property and ∑n xn is not unconditionally convergent, and we deduce that A() is meager. We also study the smallness of A() in the measure case when the Haar probability measure λ on {0, 1}? is considered. If is analytic or coanalytic, and ∑n xn is -divergent, then λ(A()) = 0 which extends the theorem of Dindo?, ?alát and Toma. Generalizing one of their examples, we show that, for every ideal on ?, with the property of long intervals, there is a divergent series of reals such that λ(A(Fin)) = 0 and λ(A()) = 1.  相似文献   

9.
10.
《Quaestiones Mathematicae》2013,36(8):985-995
Abstract

In this paper we give some equivalent conditions for the C. Huneke’s two conjectures concerning the finiteness properties of the local cohomology module , where R is a regular local ring, I is an ideal of R and i ≥ 0 is an integer.  相似文献   

11.
12.
The notion of σ?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames.  相似文献   

13.
《Quaestiones Mathematicae》2013,36(6):869-884
Abstract

This survey paper highlights a series of results in recent research on topology, geometry and categorical properties of spaces provided with a new structure in the mathematical literature, called Frölicher spaces. Without any fear of contradiction, these smooth spaces are stated to generalize the theory of differentiable manifolds. More precisely, the present study will give the state of research on this topic which historically links a fundamental theorem of calculus in so-called Boman's theorem (and its generalization) to abstract spaces, whether they are normable or not.  相似文献   

14.
15.
《Quaestiones Mathematicae》2013,36(7):977-983
Abstract

A practical number is a positive integer n such that all the positive integers mn can be written as a sum of distinct divisors of n. Let (un)n≥0 be the Lucas sequence satisfying u0 = 0, u1 = 1, and un+2 = aun+1 + bun for all integers n ≥ 0, where a and b are fixed nonzero integers. Assume a(b + 1) even and a2 + 4b > 0. Also, let be the set of all positive integers n such that |un| is a practical number. Melfi proved that is infinite. We improve this result by showing that #(x) ? x/log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding .  相似文献   

16.
A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets are specified by means of a so-called selection function, denoted by S ; these partial frames are called S-frames.

We construct free frames over S-frames using appropriate ideals, called S-ideals. Taking S-ideals gives a functor from S-frames to frames. Coupled with the functor from frames to S-frames that takes S-Lindelöf elements, it provides a category equivalence between S-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor taking S-cozero elements which is right adjoint to the functor taking S-ideals. This adjunction restricts to an equivalence of the category of completely regular S-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-? ech compactification of a completely regular S-frame, that is, its compact coreflection in the category of completely regular S-frames.

A distinguishing feature of the study of partial frames is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of frames or locales and of uniform or nearness frames. The axioms are sufficiently general to include as examples of partial frames bounded distributive lattices, σ-frames, κ-frames and frames.  相似文献   

17.
《Quaestiones Mathematicae》2013,36(7):917-936
Abstract

For a free presentation 0 τ 0 of a Leibniz algebra , the Baer invariant is called the Schur multiplier of relative to the Liezation functor or Schur Lie-multiplier. For a two-sided ideal of a Leibniz algebra , we construct a four-term exact sequence relating the Schur Lie-multipliers of and /, which is applied to study and characterize Lie-nilpotency, Lie-stem covers and Lie-capability of Leibniz algebras.  相似文献   

18.
《Quaestiones Mathematicae》2013,36(6):803-809
Abstract

In this paper, we have extended a known theorem dealing with |, pn|k summability factors to the |A, pn|k summability under weaker conditions by using an almost increasing sequence  相似文献   

19.
Themba Dube 《Order》2008,25(4):369-375
We characterise C*-quotients and C-quotients of completely regular frames in terms of ?ech-Stone compactifications and Lindelöfications, respectively. The latter is a frame-theoretic result with no spatial counterpart. We also characterise C*-quotients and dense C-quotients of completely regular frames in terms of normal covers.  相似文献   

20.
《Quaestiones Mathematicae》2013,36(8):1073-1082
Abstract

In this paper we study a two-phase population model, which distinguishes the population by two different stages

By the standard technique of characteristics, this population equation is transformed as the ordinary differential equation with nonautonomous past

where 1 ≤ p < ∞ and I = [?r, 0] (finite delay) or I = (?∞, 0] (infinite delay), E a Banach space, Φ : W1,p(I, E) → E a linear delay operator and B a nonlinear operator on E. The main result of this paper is the well-posedness of this delay equation by using the (right) multiplicative perturbation result of Desch and Schappacher in [8].  相似文献   

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