Ordinal remainders of ψ-spaces

We consider which ordinals, with the order topology, can be Stone-?ech remainders of which spaces of the form ψ(κ,M), where ω?κ is a cardinal number and M⊂ω[κ] is a maximal almost disjoint family of countable subsets of κ (MADF). The cardinality of the continuum, denoted c, and its successor cardinal, c⁺, play important roles. We show that if κ>c⁺, then no ψ(κ,M) has any ordinal as a Stone-?ech remainder. If κ?c then for every ordinal δ<κ⁺ there exists Mδ⊂ω[κ], a MADF, such that βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1. For κ=c⁺, βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1 if and only if c⁺?δ<c⁺⋅ω.     

Mrówka maximal almost disjoint families for uncountable cardinals

We consider generalizations of a well-known class of spaces, called by S. Mrówka, N∪R, where R is an infinite maximal almost disjoint family (MADF) of countable subsets of the natural numbers N. We denote these generalizations by ψ=ψ(κ,R) for κ?ω. Mrówka proved the interesting theorem that there exists an R such that |βψ(ω,R)?ψ(ω,R)|=1. In other words there is a unique free z-ultrafilter p₀ on the space ψ. We extend this result of Mrówka to uncountable cardinals. We show that for κ?c, Mrówka's MADF R can be used to produce a MADF M⊂ω[κ] such that |βψ(κ,M)?ψ(κ,M)|=1. For κ>c, and every M⊂ω[κ], it is always the case that |βψ(κ,M)?ψ(κ,M)|≠1, yet there exists a special free z-ultrafilter p on ψ(κ,M) retaining some of the properties of p₀. In particular both p and p₀ have a clopen local base in βψ (although βψ(κ,M) need not be zero-dimensional). A result for κ>c, that does not apply to p₀, is that for certain κ>c, p is a P-point in βψ.     

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].     

Fractal boundaries are not typical

Let M be a C¹n-dimensional compact submanifold of Rn. The boundary of M, ∂M, is itself a C¹ compact (n−1)-dimensional submanifold of Rn. A carefully chosen set of deformations of ∂M defines a complete subspace consisting of boundaries of compact n-dimensional submanifolds of Rn, thus the Baire Category Theorem applies to the subspace. For the typical boundary element ∂W in this space, it is the case that ∂W is simultaneously nowhere-differentiable and of Hausdorff dimension n−1.     

Locally bounded set-valued mappings and monotone countable paracompactness

We prove that the following statements are equivalent for a space X: (1) X is monotonically countably paracompact; (2) for every metric space Y there exists an operator Φ assigning to each locally bounded mapping , a locally bounded l.s.c. mapping with ?⊂Φ(?) such that Φ(?)⊂Φ(?^′) whenever ?⊂?^′, where B(Y) is the set of all non-empty closed bounded sets of Y; (3) for every metric space Y, there exist operators Φ and Ψ assigning to each u.s.c. mapping , an l.s.c. mapping and a u.s.c. mapping with ?⊂Φ(?)⊂Ψ(?) such that Φ(?)⊂Φ(?^′) and Ψ(?)⊂Ψ(?^′) whenever ?⊂?^′.     

Symmetries of two-point sets

A two-point set is a subset of the plane which meets every planar line in exactly two-points. We discuss the problem “What are the topological symmetries of a two-point set?”. Our main results assert the existence of two-point sets which are rigid and the existence of two-point sets which are invariant under the action of certain autohomeomorphism groups. We pay particular attention to the isometry group of a two-point set, and show that such groups consist only of rotations and that they may be chosen to be any subgroup of S¹ having size less than c. We also construct a subgroup of S¹ having size c that is contained in the isometry group of a two-point set.     

Rigid continua and transfinite inductive dimension

We introduce a general method of resolving first countable, compact spaces that allows accurate estimate of inductive dimensions. We apply this method to construct, inter alia, for each ordinal number α>1 of cardinality ?c, a rigid, first countable, non-metrizable continuum Sα with . Sα is the increment in some compactification of [0,1) and admits a fully closed, ring-like map onto a metric continuum. Moreover, every subcontinuum of Sα is separable. Additionally, Sα can be constructed so as to be: (1) a hereditarily indecomposable Anderson-Choquet continuum with covering dimension a given natural number n, provided α>n, (2) a hereditarily decomposable and chainable weak Cook continuum, (3) a hereditarily decomposable and chainable Cook continuum, provided α is countable, (4) a hereditarily indecomposable Cook continuum with covering dimension one, or (5) a Cook continuum with covering dimension two, provided α>2.We also produce a chainable and hereditarily decomposable space Sω(c⁺) with , , trind₀Sω(c⁺) and trInd₀Sω(c⁺) all equal to ω(c⁺), the first ordinal of cardinality c⁺.     

Choosing a sheltered middle path

We say that point x∈R² is sheltered by a continuum S⊂R² if x does not belong to the unbounded component of R²\S. Suppose that points a and b are the endpoints of each of three arcs A₀, A₁ and A₂ contained in R². We prove that there is an arc B⊂A₀∪A₁∪A₂ with its endpoints a and b such that each point of B is sheltered by the union of each two of the arcs A₀, A₁ and A₂.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.     

Non-spectral problem for a class of planar self-affine measures

The self-affine measure μM,D corresponding to an expanding matrix M∈Mn(R) and a finite subset D⊂Rn is supported on the attractor (or invariant set) of the iterated function system {?d(x)=M⁻¹(x+d)}_d∈D. The spectral and non-spectral problems on μM,D, including the spectrum-tiling problem implied in them, have received much attention in recent years. One of the non-spectral problem on μM,D is to estimate the number of orthogonal exponentials in L²(μM,D) and to find them. In the present paper we show that if a,b,c∈Z, |a|>1, |c|>1 and ac∈Z?(3Z),

     

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.     

More stratifiable function spaces

If X is a compact-covering image of a closed subspace of product of a σ-compact Polish space and a compact space, then Ck(X,M), the space of continuous maps of X into M with the compact-open topology, is stratifiable for any metric space M.If X is σ-compact Polish, K is compact and M metric then every point of Ck(X×K,M) has a closure-preserving local base, and hence this function space is M₁.     

Counting maximal topologies on countable groups and rings

Two non-discrete Hausdorff group topologies τ₁, τ₂ on a group G are called transversal if the least upper bound τ₁∨τ₂ of τ₁ and τ₂ is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits c² pairwise transversal complete group topologies on G (so c² maximal group topologies). Moreover, every abelian group G admits ²|G|² pairwise transversal complete group topologies.     

A synopsis of Combinatorial Integral Geometry

A Lie coalgebra is a coalgebra whose comultiplication Δ : M → M ? M satisfies the Lie conditions. Just as any algebra A whose multiplication ? : A ? A → A is associative gives rise to an associated Lie algebra $\text{L}$(A), so any coalgebra C whose comultiplication Δ : C → C ? C is associative gives rise to an associated Lie coalgebra $\text{L}$^c(C). The assignment C ? $\text{L}$^c(C) is functorial. A universal coenveloping coalgebra Uc(M) is defined for any Lie Lie coalgebra M by asking for a right adjoint U^c to $\text{L}$^c. This is analogous to defining a universal enveloping algebra U(L) for any Lie algebra L by asking for a left adjoint U to the functor $\text{L}$. In the case of Lie algebras, the unit (i.e., front adjunction) 1 → $\text{L}$ o U of the adjoint functor pair U ? $\text{L}$ is always injective. This follows from the Poincaré-Birkhoff-Witt theorem, and is equivalent to it in characteristic zero (x = 0). It is, therefore, natural to inquire about the counit (i.e., back adjunction) $\text{L}$^c o U^c → 1 of the adjoint functor pair $\text{L}$^c ? U^c.Theorem. For any Lie coalgebra M, the natural map$\text{L}$^c(U^cM) → M is surjective if and only if M is locally finite, (i.e., each element of M lies in a finite dimensional sub Lie coalgebra of M).An example is given of a non locally finite Lie coalgebra. The existence of such an example is surprising since any coalgebra C whose diagonal Δ is associative is necessarily locally finite by a result of that theory. The present paper concludes with a development of an analog of the Poincaré-Birkhoff-Witt theorem for Lie algebras which we choose to call the Dual Poincaré-Birkhoff-Witt Theorem and abbreviate by “The Dual PBWθ.” The constraints of the present paper, however, allow only a sketch of this theorem. A complete proof will appear in a subsequent paper. The reader may, however, consult [12], in the meantime, for details. The Dual PBWθ shows for any locally finite Lie coalgebra M the existence (in χ = 0) of a natural isomorphism of the graded Hopf algebras ₀E(U^cM) and ₀E(S^cM) associated to U^cM and to S^cM = U^c(TrivM) when U^c(M) and S^c(M) are given the Lie filtrations. [Just as U^c(M) is the analog of the enveloping algebra U(L) of a Lie algebra L, so S^c(V) is the analog of the symmetric algebra S(V) on a vector space V. Triv(M) denotes the trivial Lie coalgebra structure on the underlying vector space of M obtained by taking the comultiplication to be the zero map.]     

One more topological equivalent of CH

It is noted that CH is equivalent to the assumption that every dense pseudocompact subspace of c² contains a dense Lindelöf subspace.     

Covering dimension and finite-to-one maps

Hurewicz characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the continuous image of a zero-dimensional compact Hausdorff space by an at most n²-to-1 map.     

Geometric, topological and differentiable rigidity of submanifolds in space forms

Let M be an n-dimensional submanifold in the simply connected space form F ^n+p (c) with c + H ² > 0, where H is the mean curvature of M. We verify that if M ⁿ (n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric _M ≥ (n ? 2)(c + H ²), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or ${\mathbb{C}P^{2} \left(\frac{4}{3}(c + H^{2})\right) {\rm in} S^{7} \left(\frac{1}{\sqrt{c + H^{2}}}\right)}$ C P 2 4 3 ( c + H 2 ) in S 7 1 c + H 2 . In particular, if Ric _M > (n ? 2)(c + H ²), then M is a totally umbilic sphere. We then prove that if M ⁿ (n ≥ 4) is a compact submanifold in F ^n+p (c) with c ≥ 0, and if Ric _M > (n ? 2)(c + H ²), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

Domination by second countable spaces and Lindelöf Σ-property

Given a space M, a family of sets A of a space X is ordered by M if A={AK:K is a compact subset of M} and K⊂L implies AK⊂AL. We study the class M of spaces which have compact covers ordered by a second countable space. We prove that a space Cp(X) belongs to M if and only if it is a Lindelöf Σ-space. Under MA(ω₁), if X is compact and (X×X)\Δ has a compact cover ordered by a Polish space then X is metrizable; here Δ={(x,x):x∈X} is the diagonal of the space X. Besides, if X is a compact space of countable tightness and X²\Δ belongs to M then X is metrizable in ZFC.We also consider the class M^? of spaces X which have a compact cover F ordered by a second countable space with the additional property that, for every compact set P⊂X there exists F∈F with P⊂F. It is a ZFC result that if X is a compact space and (X×X)\Δ belongs to M^? then X is metrizable. We also establish that, under CH, if X is compact and Cp(X) belongs to M^? then X is countable.