Subsequential filters

Following N. Noble, we say that a space is subsequential if it is a subspace of a sequential space. A free filter F on ω is called subsequential if the space ω∪{F} is subsequential. In this paper, we state several properties of these filters.     

On properties of h-homogeneous spaces of first category

A metric space X is called h-homogeneous if and each nonempty open-closed subset of X is homeomorphic to X. We describe how to assign an h-homogeneous space of first category and of weight k to any strongly zero-dimensional metric space of weight ?k. We investigate the properties of such spaces. We show that if Q is the space of rational numbers and Y is a strongly zero-dimensional metric space, then Q×Yω is an h-homogeneous space and F×Q×Yω is homeomorphic to Q×Yω for any Fσ-subset F of Q×Yω. L. Keldysh proved that any two canonical elements of the Borel class α are homeomorphic. The last theorem is generalized for the nonseparable case.     

When is a Pixley-Roy hyperspace CCC?

A space X is said to satisfy condition (C) if for every Y?X with |Y|=ω₁, any family $\text{G}$ of open subsets of Y with |$\text{G}$|=ω₁ has a countable network. It is easy to see that if X satisfies condition (C), then its Pixley-Roy hyperspace $\text{F}$[X] is CCC. We show that under MA_ω1 condition (C) is also necessary for $\text{F}$[X] to be CCC, but under CH it is not.     

Positive-fraction intersection results and variations of weak epsilon-nets

A connected Finsler space (M, F) is said to be homogeneous if it admits a transitive connected Lie group G of isometries. A geodesic in a homogeneous Finsler space (G / H, F) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of G. In this paper, we study the problem of the existence of homogeneous geodesics on a homogeneous Finsler space, and prove that any homogeneous Finsler space of odd dimension admits at least one homogeneous geodesic through each point.     

The weak form of normality

A topological space is said to be paranormal if every countable discrete collection of closed sets {D _n : n < ω} can be expanded to a locally finite collection of open sets {U _n : n < ω}, i.e., D _n ? U _n and D _m ∩ U _n ≠ 0 if and only if D _m = D _n . It is proved that if F: Comp → Comp is a normal functor of degree ≥ 3 and the compact space F(X) is hereditarily paranormal, then the compact space X is metrizable.     

A first countable linearly Lindelöf not Lindelöf topological space

A topological space X is called linearly Lindelöf if every increasing open cover of X has a countable subcover. It is well known that every Lindelöf space is linearly Lindelöf. The converse implication holds only in particular cases, such as X being countably paracompact or if nw(X)<ℵω.Arhangel?skii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed ℵ₀². Consequently, a first countable linearly Lindelöf space is Lindelöf if ℵω>ℵ₀². They asked whether every linearly Lindelöf first countable space is Lindelöf in ZFC. This question is supported by the fact that all known linearly Lindelöf not Lindelöf spaces are of character at least ℵω. We answer this question in the negative by constructing a counterexample from MA+ℵω<ℵ₀².A modification of Alster?s Michael space that is first countable is presented.     

On convergence of configurations

The transition rule F of a cellular automaton may sometimes be regarded as a “rule of growth” of a crystal from a “seed” ω. A study is made of the iterates ω, Fω, F²ω,…. For certain growth rules F it is proved that when ω is “sufficiently large” the sequence F^pω “converges” to a rational polytope W. The limiting shape W depends only on F.     

o-Boundedness of free topological groups

Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F(X) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle fin_?(O,Ω) (the latter means that for every sequence 〈un〉_n∈ω of open covers of T there exists a sequence 〈vn〉_n∈ω such that vn∈[un]^<ω and for every F∈[X]^<ω there exists n∈ω with F⊂?vn). This characterization gives a consistent answer to a problem posed by C. Hernándes, D. Robbie, and M. Tkachenko in 2000.     

The property O n for finite seminormal functors

We give a finite combinatorial test for finite seminormal functors to possess the property O _n and use it in establishing that in some cases this property leads to some well-known functors. For example, if some functor F possesses the property O ₂ then F ₂ coincides with either exp₂ or the squaring functor. Hence we conclude that if F(D ^ω ₁) and D ^ω ₁ are homeomorphic then F ₂ is either exp₂ or (·)².     

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.     

A note on monotonically normal spaces

We show that a monotonically normal space X is paracompact if and only if for every increasing open cover {U _α : α < κ} of X, there is a closed cover {F _nα : n < ω, α < κ} of X such that F _nα ? U _α for n < ω, α < κ and F _nα ? F _nβ if α ≦ β.     

Metrization of Fpp-spaces

A Hausdorff space each subspace of which is a paracompact p-space is an F_pp-space. A space X is a closed hereditary Baire space if each closed subspace of X is a Baire space. Using a delicate theorem of Z. Balogh it is shown that a first-countable F_pp-space that is a closed hereditary Baire space is metrizable.     

Selective separability and its variations

A space X is said to be selectively separable (=M-separable) if for each sequence {Dn:n∈ω} of dense subsets of X, there are finite sets Fn⊂Dn (n∈ω) such that ?{Fn:n∈ω} is dense in X. On selective separability and its variations, we show the following: (1) Selective separability, R-separability and GN-separability are preserved under finite unions; (2) Assuming CH (the continuum hypothesis), there is a countable regular maximal R-separable space X such that X² is not selectively separable; (3) c{0,1} has a selectively separable, countable and dense subset S such that the group generated by S is not selectively separable. These answer some questions posed in Bella et al. (2008) [7].     

The trace-form of an algebraic number field

Let F be the rational field or a p-adic field, and let K an algebraic number field over F. If ω₁,…, ω_n is an integral basis for the ring $\text{D}$_L of integers in K, then the quadratic form Q whose matrix is (trace_K∥F(ω_iω_j)) has integral coefficients, and is called an integral trace-form. Q is determined by K up to integral equivalence. The purpose of this paper is to show that the genus of Q determines the ramification of primes in K.     

Rational power series, sequential codes and periodicity of sequences

Let R be a commutative ring. A power series f∈R[[x]] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if R is an integral extension over Z_m for some positive integer m. Let F be a field. We prove the equivalence between two versions of rationality in F[[x₁,…,x_n]]. We extend Kronecker’s criterion for rationality in F[[x]] to F[[x₁,…,x_n]]. We introduce the notion of sequential code which is a natural generalization of cyclic and even constacyclic codes over a (not necessarily finite) field. A truncation of a cyclic code over F is both left and right sequential (bisequential). We prove that the converse holds if and only if F is algebraic over F_p for some prime p. Finally, we show that all sequential codes are obtained by a simple and explicit construction.     

Parabolic Itô equations with monotone nonlinearities

In this paper the equation u_t = Lu ? F(u) + α(t, ω) is studied, where u(t) ?B₀ a Banach space. L is an unbounded self-adjoint negative definite operator. F is a monotone nonlinear potential operator. α(t, ω) is a white noise process on B₀. With suitable further restrictions on L and F it is proved that the equation has a unique solution. As t → ∞ the distribution of u(t, ω) approaches a stationary distribution which is calculated explicitly.     

On pointwise approximation of arbitrary functions by countable families of continuous functions

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf _n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F?X and every real number ε>0 one can choosen∈ω such that ∥f(x)?f_n(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A?X, there exists a continuous mapf _A:X→R ^ω such that A=f _A ^?1 (A). Splitting spaces will be studied systematically.     

On cyclic edge-connectivity of fullerenes

A graph is said to be cyclically k-edge-connected, if at least k edges must be removed to disconnect it into two components, each containing a cycle. Such a set of k edges is called a cyclic-k-edge cutset and it is called a trivial cyclic-k-edge cutset if at least one of the resulting two components induces a single k-cycle.It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclically 5-edge-connected. In this article it is shown that a fullerene F containing a nontrivial cyclic-5-edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is shown that F has a Hamilton cycle, and as a consequence at least 15·2^n/20-1/2 perfect matchings, where n is the order of F.     

Strict Embeddings of Rearrangement Invariant Spaces

A Banach space E of measurable functions on [0,1] is called rearrangement invariant if E is a Banach lattice and equimeasurable functions have identical norms. The canonical inclusion E ? F of two rearrangement invariant spaces is said to be strict if functions from the unit ball of E have absolutely equicontinuous norms in F. Necessary and sufficient conditions for the strictness of canonical inclusion for Orlicz, Lorentz, and Marcinkiewicz spaces are obtained, and the relations of this concept to the disjoint strict singularity are studied.