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.     

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.

     

Symbols of truncated Toeplitz operators

We consider three topics connected with coinvariant subspaces of the backward shift operator in Hardy spaces Hp:

-

properties of truncated Toeplitz operators;

-

Carleson-type embedding theorems for the coinvariant subspaces;

-

factorizations of pseudocontinuable functions from H¹.

These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by D. Sarason.     

Intersection cohomology of the circle actions

A classical result says that a free action of the circle S¹ on a topological space X is geometrically classified by the orbit space B and by a cohomological class e∈H²(B,Z), the Euler class. When the action is not free we have a difficult open question:

(Π)

“Is the space X determined by the orbit space B and the Euler class?”

The main result of this work is a step towards the understanding of the above question in the category of unfolded pseudomanifolds. We prove that the orbit space B and the Euler class determine:

•

the intersection cohomology of X,

•

the real homotopy type of X.

     

On the minimal members of convex expectations

In this paper, we show that for a convex expectation E[⋅] defined on L¹(Ω,F,P), the following statements are equivalent:

(i)

E is a minimal member of the set of all convex expectations defined on L¹(Ω,F,P);

(ii)

E is linear;

(iii)

two-dimensional Jensen inequality for E holds.

In addition, we prove a sandwich theorem for convex expectation and concave expectation.     

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).

     

Tiling by rectangles and alternating current

This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings by R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of this paper is an application of alternating-current circuits. The following results are obtained:

•

a necessary condition for a rectangle to be tilable by rectangles of given shapes;

•

a criterion for a rectangle to be tilable by rectangles similar to it but not all homothetic to it;

•

a criterion for a “generic” polygon to be tilable by squares.

These results generalize those of C. Freiling, R. Kenyon, M. Laczkovich, D. Rinne, and G. Szekeres.     

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.     

Bounds on sets with few distances

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered:

•

we improve the Ray-Chaudhuri-Wilson bound of the size of uniform intersecting families of subsets;

•

we refine the bound of Delsarte-Goethals-Seidel on the maximum size of spherical sets with few distances;

•

we prove a new bound on codes with few distances in the Hamming space, improving an earlier result of Delsarte.

We also find the size of maximal binary codes and maximal constant-weight codes of small length with 2 and 3 distances.     

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].     

Collapsed 5-manifolds with pinched positive sectional curvature

Let M be a closed 5-manifold of pinched curvature 0<δ?secM?1. We prove that M is homeomorphic to a spherical space form if one of the following conditions holds:

(i)

The center of the fundamental group has index ?w(δ), a constant depending on δ;

(ii)

and the fundamental group is a non-cyclic group of order ?C, a constant;

(iii)

The volume is less than ?(δ) and the fundamental group is either isomorphic to a spherical 5-space group or has an odd order, and it has a center of index ?w, a constant.

     

Noncompactness and noncompleteness in isometries of Lipschitz spaces

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip(X,E) and Lip(Y,F), for strictly convex normed spaces E and F and metric spaces X and Y:

(i)

Characterize those base spaces X and Y for which all isometries are weighted composition maps.

(ii)

Give a condition independent of base spaces under which all isometries are weighted composition maps.

(iii)

Provide the general form of an isometry, both when it is a weighted composition map and when it is not.

In particular, we prove that requirements of completeness on X and Y are not necessary when E and F are not complete, which is in sharp contrast with results known in the scalar context.     

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.     

Sequential properties of function spaces with the compact-open topology

The main results of the paper are:

(1)

If X is metrizable but not locally compact topological space, then Ck(X) contains a closed copy of S₂, and hence does not have the property AP;

(2)

For any zero-dimensional Polish X, the space Ck(X,2) is sequential if and only if X is either locally compact or the derived set X^′ is compact; and

(3)

All spaces of the form Ck(X,2), where X is a non-locally compact Polish space whose derived set is compact, are homeomorphic, and have the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n+1)st.

     

Finite Eulerian posets which are binomial, Sheffer or triangular

In this paper we study finite Eulerian posets which are binomial, Sheffer or triangular. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows:

•

We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets.

•

We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases.

•

In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets.

We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the boolean lattice by looking at smaller intervals.     

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.

     

On some contributions of John Horváth to the theory of distributions

Our main task is a presentation of J. Horváth's results concerning

•

singular and hypersingular integral operators,

•

the analytic continuation of distribution-valued meromorphic functions, and

•

a general definition of the convolution of distributions.

At some instances minor supplements to his results are given.     

h-Vectors of matroids and logarithmic concavity

Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave:

1.

The matroid complex of independent subsets of E.