Counting maximal topologies on countable groups and rings

Two non-discrete Hausdorff group topologies τ₁, τ₂ on a group G are called transversal if the least upper bound τ₁∨τ₂ of τ₁ and τ₂ is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits c² pairwise transversal complete group topologies on G (so c² maximal group topologies). Moreover, every abelian group G admits ²|G|² pairwise transversal complete group topologies.     

Minimal pseudocompact group topologies on free abelian groups

A Hausdorff topological group G is minimal if every continuous isomorphism f:G→H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence of cardinals such that

     

On the supremum of the pseudocompact group topologies

P is the class of pseudocompact Hausdorff topological groups, and P^′ is the class of groups which admit a topology T such that (G,T)∈P. It is known that every G=(G,T)∈P is totally bounded, so for G∈P^′ the supremum T^∨(G) of all pseudocompact group topologies on G and the supremum T#(G) of all totally bounded group topologies on G satisfy T^∨⊆T#.The authors conjecture for abelian G∈P^′ that T^∨=T#. That equality is established here for abelian G∈P^′ with any of these (overlapping) properties. (a) G is a torsion group; (b) |G|?c²; (c) r₀(G)=|G|=ω|G|; (d) |G| is a strong limit cardinal, and r₀(G)=|G|; (e) some topology T with (G,T)∈P satisfies w(G,T)?c; (f) some pseudocompact group topology on G is metrizable; (g) G admits a compact group topology, and r₀(G)=|G|. Furthermore, the product of finitely many abelian G∈P^′, each with the property T^∨(G)=T#(G), has the same property.     

On the density of the space of continuous and uniformly continuous functions

For X a metrizable space and (Y,ρ) a metric space, with Y pathwise connected, we compute the density of (C(X,(Y,ρ)),σ)—the space of all continuous functions from X to (Y,ρ), endowed with the supremum metric σ. Also, for (X,d) a metric space and (Y,‖⋅‖) a normed space, we compute the density of (UC((X,d),(Y,ρ)),σ) (the space of all uniformly continuous functions from (X,d) to (Y,ρ), where ρ is the metric induced on Y by ‖⋅‖). We also prove that the latter result extends only partially to the case where (Y,ρ) is an arbitrary pathwise connected metric space.To carry such an investigation out, the notions of generalized compact and generalized totally bounded metric space, introduced by the author and A. Barbati in a former paper, turn out to play a crucial rôle. Moreover, we show that the first-mentioned concept provides a precise characterization of those metrizable spaces which attain their extent.     

An explosion point space without the fixed point property

In 1988 A. Gutek proved that there exist one-point connectifications of hereditarily disconnected spaces that do not have the fixed point property. We improve on this result by constructing a one-point connectification of a totally disconnected space without the fixed point property.     

An infraconsonant nonconsonant completely regular space

Let R denote the real numbers. We construct in ZFC a countable space X such that X has exactly one non-isolated point, X is infraconsonant, and X is not consonant. We conclude that X is a completely regular space such that Isbell topology on C(X,R) is a group topology that coincides with the natural (finest splitting) topology on C(X,R), but the Isbell and compact-open topologies on C(X,R) do not coincide. The example answers two open problems in the literature.     

A new bound on the cardinality of homogeneous compacta

We show (in ZFC) that if X is a compact homogeneous Hausdorff space then |X|?²t(X), where t(X) denotes the tightness of X. It follows that under GCH the character and the tightness of such a space coincide.     

Countably compact topological group topologies on free Abelian groups from selective ultrafilters

Tkachenko showed in 1990 the existence of a countably compact group topology on the free Abelian group of size c using CH. Koszmider, Tomita and Watson showed in 2000 the existence of a countably compact group topology on the free Abelian group of size c² using a forcing model in which CH holds.Wallace's question from 1955, asks whether every both-sided cancellative countably compact semigroup is a topological group. A counterexample to Wallace's question has been called a Wallace semigroup. In 1996, Robbie and Svetlichny constructed a Wallace semigroup under CH. In the same year, Tomita constructed a Wallace semigroup from MAcountable.In this note, we show that the examples of Tkachenko, Robbie and Svetlichny, and Koszmider, Tomita and Watson can be obtained using a family of selective ultrafilters. As a corollary, the constructions presented here are compatible with the total failure of Martin's Axiom.     

Certain strict topologies on the space of regular Borel measures on locally compact groups

Let G   denote a locally compact Hausdorff group and M(G)$M(G)$ be the space of all bounded complex-valued regular Borel measures on G  . In this paper, we define two strict topologies on M(G)$M(G)$ and study various properties of these topologies such as metrizability, barrelledness and completeness. We also determine the dual space of M(G)$M(G)$ and consider various continuity properties for the convolution product on M(G)$M(G)$ under these topologies.     

Preorderable topologies and order-representability of topological spaces

We furnish characterizations of topologies that coincide with the lower topology or with the order topology of some total preorder defined on a set. Leaning on these characterizations we introduce some applications to the study of continuous and semicontinuous order-representability properties of topological spaces.     

Every scattered space is subcompact

We prove that every scattered space is hereditarily subcompact and any finite union of subcompact spaces is subcompact. It is a long-standing open problem whether every ?ech-complete space is subcompact. Moreover, it is not even known whether the complement of every countable subset of a compact space is subcompact. We prove that this is the case for linearly ordered compact spaces as well as for ω  -monolithic compact spaces. We also establish a general result for Tychonoff products of discrete spaces which implies that dense _Gδ$G_{\delta }$-subsets of Cantor cubes are subcompact.     

On a class of pseudocompact spaces derived from ring epimorphisms

A Tychonoff space X is RG if the embedding of C(X)→C(Xδ) is an epimorphism of rings. Compact RG-spaces are known and easily described. We study the pseudocompact RG-spaces. These must be scattered of finite Cantor Bendixon degree but need not be locally compact. However, under strong hypotheses, (countable compactness, or small cardinality) these spaces must, indeed, be compact. The main theorems shows, how to construct a suitable maximal almost disjoint family, and apply it to obtain examples of RG-spaces that are almost compact, locally compact, non-compact, almost-P, and of Cantor Bendixon degree 2. More complicated examples of pseudocompact non-compact RG-spaces ensue.     

Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function f∈C(Y,Z) with Y⊆Y^′ and Z⊆Z^′, find , or at least , such that ; sometimes Z=Z^′ is demanded. In this spirit the authors prove several quite general theorems in the context Y^′=κ(XI)=_∏i∈IXi in the κ-box topology (that is, with basic open sets of the form _∏i∈IUi with Ui open in Xi and with Ui≠Xi for <κ-many i∈I). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this.

Theorem. Letω?κ?α (that means: κ<α, and[β<αandλ<κ]⇒βλ<α) with α regular,be a set of non-empty spaces with eachd(Xi)<α,π[Y]=XJfor each non-emptyJ⊆Isuch that|J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets ofZ×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for everyf∈C(Y,Z)there isJ∈[I]^<αsuch thatf(x)=f(y)wheneverxJ=yJ, and (c) every such f extends to.     

On the cardinality of the θ-closed hull of sets

The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all closed neighborhoods of a point intersect C, this point is in C.     

Remainders in compactifications of topological groups

In this paper, we consider the following question: when does a topological group G have a Hausdorff compactification bG with a remainder belonging to a given class of spaces? We extend the results of A.V. Arhangel'skii by showing that if a remainder of a non-locally compact topological group G has a countable open point-network or a locally Gδ-diagonal, then G and the compactification bG of G are separable and metrizable.     

On the moduli spaces of multipolygonal linkages in the plane

One can form a polygonal linkage by identifying initial and terminal points of two free linkages. Likewise, one can form a multipolygonal linkage by identifying initial and terminal points of three free linkages. The geometric and topological properties of moduli spaces of multipolygonal linkages in the plane are studied. These spaces are compact algebraic varieties. Some conditions under which these spaces are smooth manifolds, cross products or disjoint unions of moduli spaces of polygonal linkages, or connected, are determined. Dimensions in smooth manifold cases and some Euler characteristics are computed. A classification of generic multiquadrilateral linkages is also made.     

Solution of Shapirovskii's question

Boris Shapirovskii posed the following question: “Suppose a continuous mapping from one compact space onto another is given. Suppose that the π-character of any point in the domain is greater than the weight of the target space. Will there be two disjoint closed sets in the domain mapping onto?”. There is a zero-dimensional counterexample (Corollary 12), but the one cardinal up version is true in zero-dimensional case (Corollary 14). Nevertheless, there always will be two disjoint open sets mapping densely (Theorem 7).