A note on a theorem of Talagrand

We provide an elementary argument to show that if for a hemicompact kR-space X the space Cp(X) contains a subset S which separates the points of X and is dominated by irrationals, i.e. S is covered by a family of compact sets such that Kα⊂Kβ for α?β, then Cp(X) is also dominated by irrationals; consequently Cp(X) is K-analytic. This fact (which fails for non-hemicompact spaces X) extends an old result of Talagrand.     

An n-dimensional version of Steinhaus' chessboard theorem

The main result of this paper is an n-dimensional version of the Steinhaus' chessboard theorem. Our theorem implies the Poincaré theorem as well as its parametric extension. But it is known that the Poincaré theorem is equivalent to the Brouwer Fixed-Point theorem.     

Limit sets and the Poincaré-Bendixson Theorem in impulsive semidynamical systems

We consider semidynamical systems with impulse effects at variable times and we discuss some properties of the limit sets of orbits of these systems such as invariancy, compactness and connectedness. As a consequence we obtain a version of the Poincaré-Bendixson Theorem for impulsive semidynamical systems.     

Topological multiple recurrence for polynomial configurations in nilpotent groups

We establish a general multiple recurrence theorem for an action of a nilpotent group by homeomorphisms of a compact space. This theorem can be viewed as a nilpotent version of our recent polynomial Hales-Jewett theorem (Ann. Math. 150 (1999) 33) and contains nilpotent extensions of many known “abelian” results as special cases.     

A probabilistic version of a theorem of Kegel

Let G be a finite group and α an automorphism of G of prime order p. It is shown that if the elements x of G satisfy the equation with sufficiently high probability (depending only on p) then G is nilpotent.Received: 8 January 2005     

A fixed point theorem for the pseudo-circle

Let f:C→C be a self-map of the pseudo-circle C. Suppose that C is embedded into an annulus A, so that it separates the two components of the boundary of A. Let F:A→A be an extension of f to A (i.e. F|C=f). If F is of degree d then f has at least |d−1| fixed points. This result generalizes to all plane separating circle-like continua.     

A relative version of Webb’s theorem on the equivariant chain complex of a subgroup complex

We prove a relative version of the theorem of Webb that the augmented chain complex of the p-subgroup complex of a finite group, considered as a complex of modules for the group, is homotopy equivalent to a complex of projectives. This allows us to take into account the group of automorphisms of the group.     

Splitting theorem, Poincaré-Hopf theorem and jumping nonlinear problems

In this paper, we establish a Gromoll-Meyer splitting theorem and a shifting theorem for J∈C^2-0(E,R) and by using the finite-dimensional approximation, mollifiers and Morse theory we generalize the Poincaré-Hopf theorem to J∈C¹(E,R) case. By combining the Poincaré-Hopf theorem and the splitting theorem, we study the existence of multiple solutions for jumping nonlinear elliptic equations.     

Paracompactness of paratopological groups which are GO-spaces

Let G be a paratopological group which is a GO-space. We have showed that if the multiplication in G preserves the order on G, then G is paracompact.     

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.

     

On a theorem of Rhemtulla on polycyclic groups

Let H be a subgroup of the polycyclic-by-finite group G and denote the automorphism group of G by Γ. We prove that there exists an integer d such that in the poset {?_γ∈∑H^γ:∑ a subset of Γ} of all intersections of images H^γ of H under Γ, chains have length at most d. In particular the poset satisfies the minimal condition. This extends and improves a theorem of A.H. Rhemtulla. We also provide a very different proof of Rhemtulla’s theorem.     

A note on a modified version of the Poincaré-Birkhoff theorem

We show, by constructing a suitable area preserving homeomorphism without fixed points, that the twist condition considered at zero and at infinity in a recent modified version of the Poincaré-Birkhoff fixed point theorem cannot be interchanged.     

Abelian groups admitting a Fréchet-Urysohn pseudocompact topological group topology

We show that every Abelian group G with r₀(G)=|G|=|G|^ω admits a pseudocompact Hausdorff topological group topology T such that the space (G,T) is Fréchet-Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet-Urysohn space if for every prime divisor p of n and every integer k≥0, the Ulm-Kaplansky invariant f_p,k of G satisfies (f_p,k)^ω=f_p,k provided that f_p,k is infinite and f_p,k>f_p,i for each i>k.Our approach is based on an appropriate dense embedding of a group G into a Σ-product of circle groups or finite cyclic groups.     

An asymmetric Ellis theorem

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, Tk, to obtain the following asymmetric Ellis theorem which applies to the example above:Whenever (X,⋅,T) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both (X,⋅,T) and (X,⋅,Tk), and inversion is a homeomorphism between (X,T) and (X,Tk).This generalizes the classical Ellis theorem, because T=Tk when (X,T) is locally compact Hausdorff.     

A new version of the Hahn-Banach theorem

We discuss a new version of the Hahn-Banach theorem, with applications to linear and nonlinear functional analysis, convex analysis, and the theory of monotone multifunctions. We show how our result can be used to prove a localized version of the Fenchel-Moreau formula - even when the classical Fenchel-Moreau formula is valid, the proof of it given here avoids the problem of the vertical hyperplane. We give a short proof of Rockafellars fundamental result on dual problems and Lagrangians - obtaining a necessary and sufficient condition instead of the more usual sufficient condition. We show how our result leads to a proof of the (well-known) result that if a monotone multifunction on a normed space has bounded range then it has full domain. We also show how our result leads to generalizations of an existence theorem with no a priori scalar bound that has proved very useful in the investigation of monotone multifunctions, and show how the estimates obtained can be applied to Rockafellars surjectivity theorem for maximal monotone multifunctions in reflexive Banach spaces. Finally, we show how our result leads easily to a result on convex functions that can be used to establish a minimax theorem.     

A categorical approach to the maximum theorem

Berge's maximum theorem gives conditions ensuring the continuity of an optimised function as a parameter changes. In this paper we state and prove the maximum theorem in terms of the theory of monoidal topology and the theory of double categories.This approach allows us to generalise (the main assertion of) the maximum theorem, which is classically stated for topological spaces, to pseudotopological spaces and pretopological spaces, as well as to closure spaces, approach spaces and probabilistic approach spaces, amongst others. As a part of this we prove a generalisation of the extreme value theorem.     

FLUIDIFICATION OF QUIVERS AND THE TIETZE EXTENSION THEOREM FOR DIAGRAMS OF TOPOLOGICAL SPACES

For quivers with no directed cycles, the concept of fluidification is defined and used to prove that, under corresponding assumptions, the Tietze extension theorem for diagrams of topological spaces modeled on such quivers holds. Another application of the given concept of fluidification -is a characterization of certain diagrams of proper maps.     

A non-linear version of Hunt-Lion's theorem from the point of view of T-accretivity

In the classical topological context, Dellacherie [10] has given a non-linear version of Hunt's theorem characterizing the proper kernels verifying the complete maximum principle as those closing a submarkovian resolvent. In this paper we study the relation between this non-linear version of Hunt's theorem and T-accretivity.     

The hyperspace of hereditarily decomposable subcontinua of a cube is the Hurewicz set

The hyperspaces of hereditarily decomposable continua and of decomposable subcontinua without pseudoarcs in the cube of dimension greater than 2 are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0,1]. Moreover, in such a cube, all indecomposable subcontinua form a homotopy dense subset of the hyperspace of (nonempty) subcontinua.     

Combinations not tolerated by normal topological groups

We show that if X is not paracompact then one can find a compact space K such that X⊕K does not embed closedly into any normal topological group.