A thin-tall Boolean algebra which is isomorphic to each of its uncountable subalgebras

Theorem A ℵ_1?. There is a Boolean algebra B with the following properties:

(1)

B is thin-tall, and

(2)

B is downward-categorical.

That is, every uncountable subalgebra of B is isomorphic to B.     

Extending compact topologies to compact Hausdorff topologies in ZF

Within the framework of Zermelo-Fraenkel set theory ZF, we investigate the set-theoretical strength of the following statements:

(1)

For every family(Ai)_i∈Iof sets there exists a family(Ti)_i∈Isuch that for everyi∈I(Ai,Ti)is a compactT₂space.

(2)

For every family(Ai)_i∈Iof sets there exists a family(Ti)_i∈Isuch that for everyi∈I(Ai,Ti)is a compact, scattered, T₂space.

(3)

For every set X, every compactR₁topology (itsT₀-reflection isT₂) on X can be enlarged to a compactT₂topology.

We show:

(a)

(1) implies every infinite set can be split into two infinite sets.

(b)

(2) iff AC.

(c)

(3) and “there exists a free ultrafilter” iff AC.

We also show that if the topology of certain compact T₁ spaces can be enlarged to a compact T₂ topology then (1) holds true. But in general, compact T₁ topologies do not extend to compact T₂ ones.     

Bockstein basis and resolution theorems in extension theory

We prove a generalization of the Edwards-Walsh Resolution Theorem:

Theorem. Let G be an abelian group withPG=P, where. Letn∈Nand let K be a connected CW-complex withπn(K)≅G,πk(K)≅0for0?k<n. Then for every compact metrizable space X with XτK (i.e., with K an absolute extensor for X), there exists a compact metrizable space Z and a surjective mapπ:Z→Xsuch that

(a)

π is cell-like,

(b)

dimZ?n, and

(c)

ZτK.

     

The Fréchet-Urysohn property of Pixley-Roy hyperspaces

Let F[X] be the Pixley-Roy hyperspace of a regular space X. In this paper, we prove the following theorem.

Theorem. For a space X, the following are equivalent:

(1)

F[X]is a k-space;

(2)

F[X]is sequential;

(3)

F[X]is Fréchet-Urysohn;

(4)

Every finite power of X is Fréchet-Urysohn for finite sets;

(5)

Every finite power ofF[X]is Fréchet-Urysohn for finite sets.

As an application, we improve a metrization theorem onF[X].     

Rothberger bounded groups and Ramsey theory

We show that:

(1)

Rothberger bounded subgroups of σ-compact groups are characterized by Ramseyan partition relations (Corollary 4).

(2)

For each uncountable cardinal κ there is a T₀ topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is not a closed subspace of any σ-compact space (Theorem 8).

(3)

For each uncountable cardinal κ there is a T₀ topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is σ-compact (Corollary 17).

     

Properties of the Holmes space

The following properties of the Holmes space H are established:

(i)

H has the Metric Approximation Property (MAP).

(ii)

The w^∗-closure of the set of extreme points of the unit ball BH^∗ of the dual space H^∗ is the whole ball BH^∗.

A family of compact subsets X⊂U of the Urysohn space is described such that the Lipschitz-free space F(X) has a finite-dimensional decomposition and is not complemented in H.     

Spaces with large weak extent

In this paper, we show the following statements:

(1)

For any cardinal κ, there exists a pseudocompact centered-Lindelöf Tychonoff space X such that we(X)?κ.

(2)

Assuming ℵ₀²=ℵ₁², there exists a centered-Lindelöf normal space X such that we(X)?ω₁.

     

Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

(1)

A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;

(2)

A poset is continuous iff its Scott topology is completely distributive;

(3)

A topological T₀ space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;

(4)

A topological T₁ space is a discrete space iff its topology is completely distributive.

These results generalize the relevant results obtained by J.D. Lawson for dcpos.     

The topology of continua that are approximated by disjoint subcontinua

Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:

(1)

X is non-Suslinean.

(2)

If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.

(3)

If X is G-like, then X is indecomposable.

(4)

If all lie in the same ray and X is finitely cyclic, then X is indecomposable.

     

Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:

•

T contains all weakly Lindelöf Banach spaces;

•

l^∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30-42], pp. 30-31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l^∞/c₀)∉T.

•

T is stable under weak homeomorphisms;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.

     

Normality and countable paracompactness of hyperspaces of ordinals

For an ordinal α, α² denotes the collection of all nonempty closed sets of α with the Vietoris topology and K(α) denotes the collection of all nonempty compact sets of α with the subspace topology of α². It is well known that α² is normal iff cfα=1. In this paper, we will prove that for every nonzero-ordinal α:

(1)

α² is countably paracompact iff cfα≠ω.

(2)

K(α) is countably paracompact.

(3)

K(α) is normal iff, if cfα is uncountable, then cfα=α.

In (3), we use elementary submodel techniques.     

D-spaces and thick covers

The following results are obtained.

-

An open neighbornet U of X has a closed discrete kernel if X has an almost thick cover by countably U-close sets.

-

Every hereditarily thickly covered space is aD and linearly D.

-

Every t-metrizable space is a D-space.

-

X is a D-space if X has a cover {Xα:α<λ} by D-subspaces such that, for each β<λ, the set ?{Xα:α<β} is closed.

     

Definable sets of generators in maximal cofinitary groups

A group G?Sym(N) is cofinitary if g has finitely many fixed points for every g∈G except the identity element. In this paper, we discuss the definability of maximal cofinitary groups and some related structures. More precisely, we show the following two results:

(1)

Assuming V=L, there is a set of permutations on N which generates a maximal cofinitary group.

(2)

Assuming V=L, there is a mad permutation family in Sym(N).

     

Ubiquity of odometers in topological dynamical systems

Let M be the Cantor space or an n-manifold with C(M,M) the set of continuous self-maps of M. We prove the following:

(1)

There is a residual set of points (x,f) in M×C(M,M) all of which generate as their ω-limit set a particular, unique adding machine.

(2)

Moreover, if M has the fixed point property, then a generic f∈C(M,M) generates uncountably many distinct copies of every possible adding machine.

     

Function spaces from core compact coherent spaces to continuous B-domains

It is proved in this paper that for a continuous B-domain L, the function space [X→L] is continuous for each core compact and coherent space X. Further, applications are given. It is proved that:

(1)

the function space from the unit interval to any bifinite domain which is not an L-domain is not Lawson compact;

(2)

the Isbell and Scott topologies on [X→L] agree for each continuous B-domain L and core compact coherent space X.

     

The space of adding machines generated by continuous self maps of manifolds

Let M be the Cantor space or an n-dimensional manifold with C(M,M) the set of continuous self-maps of M, and . We prove the following:

(1)

If α≠∞, then Sα(M) is a nowhere dense subset of M×C(M,M) that contains no isolated points.

(2)

If α?β, then .

     

Hyponormal operators with rank-two self-commutators

In this paper it is shown that if T∈L(H) satisfies

(i)

T is a pure hyponormal operator;

(ii)

[T^∗,T] is of rank two; and

(iii)

ker[T^∗,T] is invariant for T,

then T is either a subnormal operator or the Putinar's matricial model of rank two. More precisely, if T_|ker[T^∗,T] has a rank-one self-commutator then T is subnormal and if instead T_|ker[T^∗,T] has a rank-two self-commutator then T is either a subnormal operator or the kth minimal partially normal extension, , of a (k+1)-hyponormal operator Tk which has a rank-two self-commutator for any k∈Z₊. Hence, in particular, every weakly subnormal (or 2-hyponormal) operator with a rank-two self-commutator is either a subnormal operator or a finite rank perturbation of a k-hyponormal operator for any k∈Z₊.     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.