Glicksberg's Theorem Research Articles

Tychonoff spaces and a ring theoretic order on C(X)

The reduced ring order (rr-order) is a natural partial order on a reduced ring R given by r≤rrs if r2=rs. It can be studied algebraically or topologically in rings of the form C(X). The focus here is on those reduced rings in which each pair of elements has an infimum in the rr-order, and what this implies for X. A space X is called rr-good if C(X) has this property. Surprisingly both locally connected and basically disconnected spaces share this property. The rr-good property is studied under various topological conditions including its behaviour under Cartesian products. The product of two rr-good spaces can fail to be rr-good (e.g., βR×βR), however, the product of a P-space and an rr-good weakly Lindelöf space is always rr-good. P-spaces, F-spaces and U-spaces play a role, as do Glicksberg's theorem and work by Comfort, Hindman and Negrepontis.

Hausdorff compactifications in ZF

For a compactification αX of a Tychonoff space X, the algebra of all functions f∈C(X) that are continuously extendable over αX is denoted by Cα(X). It is shown that, in a model of ZF, it may happen that a discrete space X can have non-equivalent Hausdorff compactifications αX and γX such that Cα(X)=Cγ(X). Amorphous sets are applied to a proof that Glicksberg's theorem that βX×βY is the Čech-Stone compactification of X×Y when X×Y is a Tychonoff pseudocompact space is false in some models of ZF. It is noticed that if all Tychonoff compactifications of locally compact spaces had C⁎-embedded remainders, then van Douwen's choice principle would be satisfied. Necessary and sufficient conditions for a set of continuous bounded real functions on a Tychonoff space X to generate a compactification of X are given in ZF. A concept of a maximal functional compactification is investigated in the absence of the axiom of choice.

Amenability and Fan–Glicksberg theorem for set-valued mappings

In this paper, we begin by discussion of some well known results on the existence of left invariant means in the spaces: LUC(S), AP(S) and W AP(S) with Hahn-Banach extension theorem. We then give a new and precise proof of the well known Fan–Glicksberg fixed point theorem. This is then followed by a discussion on some related open problems.

The Glicksberg theorem for Tychonoff extension properties

It is a well known result of Glicksberg that the Stone–Čech compactification can be factorized in a product of Tychonoff spaces iff the product of the spaces is pseudocompact. In this paper we present some Glicksberg-like results for Tychonoff extension properties. In particular, specific theorems for certain compact-like properties are established. Furthermore, we redefine the notion of power of an ultrafilter of N⁎, originally introduced by Booth, and establish a linear order for these powers.

On the Glicksberg theorem for locally quasi-convex Schwartz groups

On the Glicksberg theorem for locally quasi-convex Schwartz groups

Dieudonné completion and [formula omitted]-group actions

If G ( X ) denotes either the free topological group or the free Abelian topological group over a topological space X, we prove that ∏ i = 1 n G ( X i ) is a hemibounded b f -group whenever each X i is a pseudocompact space (which provides a new way to generate this kind of topological groups), and we show that the equality μ ( X × ∏ i = 1 n G ( X i ) ) = μ X × ∏ i = 1 n G ( β X i ) holds whenever X is a hemibounded b f -space (where μY stands for the Dieudonné completion of Y). By means of the Dieudonné completion we prove that every pseudocompact space X is G-Tychonoff whenever G is a b f -group and that the maximal G-compactification of X coincides with βX. We apply this result to obtain a partial version for G-spaces of Glicksberg's theorem on pseudocompactness and we analyze when the maximal G-compactification of a G-space X coincides with the Stone–Čech compactification of X in the case when G is a metrizable group.

On the existence of symmetric mixed strategy equilibria

In this paper we provide a result ensuring the existence of symmetric mixed strategy equilibria in symmetric games. We apply the fixed point theorem of Glicksberg and Fan analogously to the way in which Moulin [Moulin, H., 1986. Game Theory for the Social Sciences, 2nd Edition. New York University Press, New York.] uses Kakutani's theorem to prove the existence of a symmetric equilibrium in pure strategies.

Totally Bounded Quasi-Uniformities and Pseudocompactness of Bispaces

We characterize pairwise Tychonoff bispaces that admit only totally bounded quasi-uniformities in terms of a suitable notion of bitopological pseudocompactness. We also show that a pairwise Tychonoff bispace has a unique (up to equivalence) bicompactification if and only if it admits a unique totally bounded quasi-unifomity. These results extend classical theorems of R. Doss for uniform spaces to the quasi-uniform (bitopological) setting, and are applied to the study of T0 topological spaces that admit a unique quasi-uniformity and a unique bicompactification, respectively. Finally, we discuss the problem of extending the classical Glicksberg theorem on product of pseudocompact spaces to bispaces and a partial solution is obtained.

Some remarks on the product of two C α-compact subsets

For a cardinal α, we say that a subset B of a space X is C α-compact in X if for every continuous function $$f:X \to \mathbb{R}^\alpha ,f\left[ B \right]$$ is a compact subset of $$\mathbb{R}^\alpha $$ . If B is a C-compact subset of a space X, then ϱ(B, X) denotes the degree of C α-compactness of B in X. A space X is called α-pseudocompact if X is C α-compact into itself. For each cardinal α, we give an example of an α-pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in “Some cardinal generalizations of pseudocompactness” Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.

Weakly pseudocompact subsets of nuclear groups

Let G be an Abelian topological group and G + the group G endowed with the weak topology induced by continuous characters. We say that G respects compactness (pseudocompactness, countable compactness, functional boundedness) if G and G + have the same compact (pseudocompact, countably compact, functionally bounded) sets. The well-known theorem of Glicksberg that LCA groups respect compactness was extended by Trigos-Arrieta to pseudocompactness and functional boundedness. In this paper we generalize these results to arbitrary nuclear groups, a class of Abelian topological groups which contains LCA groups and nuclear locally convex spaces and is closed with respect to subgroups, separated quotients and arbitrary products.

Maximally Almost Periodic Groups and a Theorem of Glicksberga

ABSTRACT:It is well known that if G is a locally compact Abelian group (LCA group) with Bohr compactification (β(G), σ) then σ(G) is normal in β(G) and, by a beautiful theorem of Glicksberg, we have that A⊂σ(G) is compact if and only if σ–1(A) ⊂G is compact. The aim of this paper is to study maximally almost periodic (MAP) groups which have these properties and the results obtained are as follows. (1) If G is a σ‐compact locally compact MAP group with Bohr compactification (β(G), σ) and σ(G) is normal in β(G), then for each gεβ(G), the automorphism induced by σ and conjugation by g is actually a topological isomorphism. (2) A finite extension of a LCA group is a MAP group and it has the property that A⊂σ(G) is compact if and only if σ–1(A) ⊂G is compact, and (3) A discrete MAP group G with Bohr compactification (β(G), σ) satisfying both of the properties being considered must be Abelian by finite, i.e., a finite extension of an Abelian group.

The Glicksberg Theorem on Weakly Compact Sets for Nuclear Groups

ABSTRACTBy the weak topology on an Abelian topological group we mean the topology induced by the family of all continuous characters. A well‐known theorem of I. Glicksberg says that weakly compact subsets of locally compact Abelian (LCA) groups are compact. D. Remus and F.J. Trigos‐Arrieta [1993. Proceedings Amer. Math. Soc. 117] observed that Glicksberg's theorem remains valid for closed subgroups of any product of LCA groups. Here we show that, in fact, it remains valid for all nuclear groups, a class of Abelian topological groups introduced by the first author in the monograph, “Additive subgroups of topological vector spaces” [1991. Lecture Notes in Math. 1466].

Generating subsemigroups, orders, and a theorem of Glicksberg

Generating subsemigroups, orders, and a theorem of Glicksberg

Uniformities on a Product

All topological spaces shall be uniformizable (completely regular Hausdorff). A uniformity on X shall be viewed as a collection μ of coverings of X, via the manner of Tukey [20] and Isbell [16], and the associated uniform space denoted μX. Given the uniformizable topological space X, we shall be concerned with compatible uniformities as follows (discussed more carefully in § 1). The fine uniformity α (finest compatible with the topology); the “cardinal reflections“ αm of α (m an infinite cardinal number) ; αc, the weak uniformity generated by the real-valued continuous functions.With μ standing, generically, for one of these uniformities, we consider the question: when is μ(X × Y) = μX × μY For μ = αℵ0 (the finest compatible precompact uniformity), the problem is equivalent to that of whenβ(X × Y) = βX × βY,β denoting Stone-Cech compactification; this is answered by the theorem of Glicksberg [9]. For μ = α, we have Isbell's generalization [16, VI1.32].

On a theorem of Glicksberg and fixed point properties of semigroups

On a theorem of Glicksberg and fixed point properties of semigroups

Hewitt Realcompactifications of Products

The Hewitt realcompactification vX of a completely regular Hausdorff space X has been widely investigated since its introduction by Hewitt [17]. An important open question in the theory concerns when the equality v(X × Y) = vX × vY is valid. Glicksberg [10] settled the analogous question in the parallel theory of Stone-Čech compactifications: for infinite spaces X and Y, β(X × Y) = βX × β Y if and only if the product X × Y is pseudocompact. Work of others, notably Comfort [3; 4] and Hager [13], makes it seem likely that Glicksberg's theorem has no equally specific analogue for v(X × Y) = vX × vY. In the absence of such a general result, particular instances may tend to be attacked by ad hoc techniques resulting in much duplication of effort.