Ramsey-Milman phenomenon,Urysohn metric spaces,and extremely amenable groups

In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso of isometries of Urysohn’s universal complete separable metric space , equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if is replaced with any generalized Urysohn metric spaceU that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into a topological group of the form Iso(U), our result implies that every topological group embeds into an extremely amenable group (one admitting an invariant multiplicative mean on bounded right uniformly continuous functions). By way of the proof, we show that every topological group is approximated by finite groups in a certain weak sense. Our technique also results in a new proof of the extreme amenability (fixed point on compacta property) for infinite orthogonal groups. Going in the opposite direction, we deduce some Ramsey-type theorems for metric subspaces of Hilbert spaces and for spherical metric spaces from existing results on extreme amenability of infinite unitary groups and groups of isometries of Hilbert spaces.     

Cut-point spaces

The notion of a cut-point space is introduced as a connected topological space without any non-cut point. It is shown that a cut-point space is infinite. The non-cut point existence theorem is proved for general (not necessarily ) topological spaces to show that a cut-point space is non-compact. Also, the class of irreducible cut-point spaces is studied and it is shown that this class (up to homeomorphism) has exactly one member: the Khalimsky line.

     

A class of bornological barrelled spaces which are not ultrabornological

N. Bourbaki, [1, p. 35], notices that it is not known if every bornological barrelled space is ultrabornological. In this paper we prove that if E is the topological product of an infinite family of bornological barrelled spaces, of non-zero dimension, there exists an infinite number of bornological barrelled subspaces ofE, which are not ultrabornological. We also give some examples of barrelled normable spaces which are not ultrabornological.Supported in part by the Patronato para el Fomento de la Investigación en la Universidad.     

Minimal embeddings of topological spaces into the real line

A theorem describing ?-minimal topological spaces is proved. These are spaces (X, τ) topologically embeddable into the real line ? and not possessing this property under the replacement of τ by a weaker topology.     

Scott's rigidity theorem for Seifert fibered spaces; revisited

We will present a new proof of the rigidity theorem for Seifert fibered spaces of infinite by Scott (1983) in the case when the base of the fibration is a hyperbolic triangle 2-orbifold. Our proof is based on arguments in the rigidity theorem for hyperbolic 3-manifolds by Gabai (1997).

     

Injective spaces via adjunction

The work of the present author and his coauthors over the past years gives evidence that it may be useful to regard each topological space as a kind of enriched category, by interpreting the convergence relation x→x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from enriched Category Theory for the investigation of topological spaces. Topological theories introduced by the author provide a convenient general setting for appropriately transferring these concepts and ideas to the world of topological spaces and some other geometric objects such as approach spaces. Using tools like adjunction and the Yoneda lemma, we show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on . This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.     

Baire sets in topological spaces

Summary In this paper we study topological properties of Baire sets in various classes of spaces. The main results state that a Baire set in a realcompact space is realcompact; a Baire set in a topologically complete space is topologically complete; and that a pseudocompact Baire set in any topological space is a zero-set. As a consequence, we obtain new characterizations of realcompact and pseudocompact spaces in terms of Baire sets of their Stone-ech compactifications. (Lorch in [3] using a different method has obtained either implicitly or explicitly the same results concerning Baire sets in realcompact spaces.) The basic tools used for these proofs are first, the notions of anr-compactification andr-embedding (see below for definitions), which have also been defined and used independently byMrówka in [4]; second, the idea included in the proof of the theorem: Every compact Baire set is aG as given inHalmos' text on measure theory [2; Section 51, theorem D].The author wishes to thank Professor W. W.Comfort for his valuable advice in the preparation of this paper.     

Epi-topology on spaces of lower semicontinuous functions into partially ordered spaces and hyperspaces

The concept of lower semicontinuity is extended to functions mapping into partially ordered spaces. A study is made of spaces of such lower semicontinuous functions under the epi-topology. These spaces are subspaces of hyperspaces with the Fell topology. The closure of such a function space in the hyperspace is characterized for certain spaces. A continuous selection theorem is established, showing that most such function spaces are not ech-complete.     

The enriched Vietoris monad on representable spaces

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “up-set monad” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of Kock–Zöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “up-set monad”. We emphasise that these generic categorical notions and results can be indeed connected to more “classical” topology: for topological spaces, the “up-set monad” becomes the lower Vietoris monad, and the statement “X   is totally cocomplete if and only if X^op$X^{\mathrm{op}}$ is totally complete” specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces as precisely the continuous lattices.     

Coarse embeddings of metric spaces into Banach spaces

There are several characterizations of coarse embeddability of locally finite metric spaces into a Hilbert space. In this note we give such characterizations for general metric spaces. By applying these results to the spaces , we get their coarse embeddability into a Hilbert space for . This together with a theorem by Banach and Mazur yields that coarse embeddability into and into are equivalent when . A theorem by G.Yu and the above allow us to extend to , , the range of spaces, coarse embeddings into which is guaranteed for a finitely generated group to satisfy the Novikov Conjecture.

     

Measures on product spaces and the existence of strong Baire liftings

Some notions are introduced for studying measures on product spaces, the main concept being that of property (*). In case when the topological factors are separable metric spaces, this property is equivalent to the completion regularity. We prove that (*) is preserved under arbitrary products of measure spaces. As a consequence, we deduce a series of related results in measure theory (some of which are known). In particular, the following extension of a result by Losert is obtained: Subject to CH, every product of ₂ many completion regular measures, each supported on any product of ₁ many compact metric spaces admits a strong Baire lifting.     

Geometric cohomology frames on Hausmann-Holm-Puppe conjugation spaces

For certain manifolds with an involution the mod 2 cohomology ring of the set of fixed points is isomorphic to the cohomology ring of the manifold, up to dividing the degrees by two. Examples include complex projective spaces and Grassmannians with the standard antiholomorphic involution (with real projective spaces and Grassmannians as fixed point sets).

Hausmann, Holm and Puppe have put this observation in the framework of equivariant cohomology, and come up with the concept of conjugation spaces, where the ring homomorphisms arise naturally from the existence of what they call cohomology frames. Much earlier, Borel and Haefliger had studied the degree-halving isomorphism between the cohomology rings of complex and real projective spaces and Grassmannians using the theory of complex and real analytic cycles and cycle maps into cohomology.

The main result in the present note gives a (purely topological) connection between these two results and provides a geometric intuition into the concept of a cohomology frame. In particular, we see that if every cohomology class on a manifold with involution is the Thom class of an equivariant topological cycle of codimension twice the codimension of its fixed points (inside the fixed point set of ), these topological cycles will give rise to a cohomology frame.

     

Fuzzy topological spaces and fuzzy compactness

It is the purpose of this paper to go somewhat deeper into the structure of fuzzy topological spaces. In doing so we found we had to alter the definition of a fuzzy topology used up to now. We shall also introduce two functors $\text{}\text{?}$gw and $\text{}\text{?}$gi which will allow us to see more clearly the connection between fuzzy topological spaces and topological spaces. Finally we shall introduce the concept of fuzzy compactness as the generalization of compactness in topology. It will be shown in a following publication that contrary to the results obtained up to now, the Tychonoff-product theorem is safeguarded with fuzzy compactness.     

On a new generalization of metric spaces

In this paper, we introduce the \({\mathcal {F}}\)-metric space concept, which generalizes the metric space notion. We define a natural topology \(\tau _{{\mathcal {F}}}\) in such spaces and we study their topological properties. Moreover, we establish a new version of the Banach contraction principle in the setting of \({\mathcal {F}}\)-metric spaces. Several examples are presented to illustrate our study.     

Morse-Palais lemma for nonsmooth functionals on normed spaces

Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in Banach spaces, our result does not require the -smoothness of functions nor the completeness of spaces. Therefore it is stronger than the classical one but its proof is very simple.

     

On C‐Suslin spaces

We prove a closed graph theorem for Baire locally convex spaces (for Baire linear topological spaces) in the domain and weakly C‐Suslin locally convex spaces (respectively, for C‐Suslin linear topological spaces) in the range which improves some classic closed graph theorems and other, more recent, related results.     

Hindman spaces

A topological space is Hindman if for every sequence in there exists an infinite so that the sequence , indexed by all finite sums over , is IP-converging in . Not all sequentially compact spaces are Hindman. The product of two Hindman spaces is Hindman.

Furstenberg and Weiss proved that all compact metric spaces are Hindman. We show that every Hausdorff space that satisfies the following condition is Hindman:

Consequently, there exist nonmetrizable and noncompact Hindman spaces. The following is a particular consequence of the main result: every bounded sequence of monotone (not necessarily continuous) real functions on has an IP-converging subsequences.

     

3-connected planar spaces uniquely embed in the sphere

We characterize those locally connected subsets of the sphere that have a unique embedding in the sphere -- i.e., those for which every homeomorphism of the subset to itself extends to a homeomorphism of the sphere. This implies that if is the closure of an embedding of a 3-connected graph in the sphere such that every 1-way infinite path in has a unique accumulation point in , then has a unique embedding in the sphere. In particular, the standard (or Freudenthal) compactification of a 3-connected planar graph embeds uniquely in the sphere.