Classifying homeomorphism groups of infinite graphs

In this paper, we classify topologically the homeomorphism groups H(Γ) of infinite graphs Γ with respect to the compact-open and the Whitney topologies.     

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.     

A topological characterization of LF-spaces

We present a topological characterization of LF-spaces and detect small box-products that are (locally) homeomorphic to LF-spaces.     

A new Lindelöf topological group

We show that the subsemigroup of the product of ω₁-many circles generated by the L-space constructed by J. Moore is again an L-space. This leads to a new example of a Lindelöf topological group. The question whether all finite powers of this group are Lindelöf remains open.     

Embedding paratopological groups into topological products

We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the Hausdorff number of G is countable, i.e., for every neighbourhood U of the neutral element e of G there exists a countable family γ of neighbourhoods of e such that _?V∈γVV−1⊆U. Similarly, we prove that a regular paratopological group G can be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the index of regularity of G is countable.As a by-product, we show that a regular totally ω-narrow paratopological group with countable index of regularity is Tychonoff.     

o-Boundedness of free topological groups

Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F(X) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle fin_?(O,Ω) (the latter means that for every sequence 〈un〉_n∈ω of open covers of T there exists a sequence 〈vn〉_n∈ω such that vn∈[un]^<ω and for every F∈[X]^<ω there exists n∈ω with F⊂?vn). This characterization gives a consistent answer to a problem posed by C. Hernándes, D. Robbie, and M. Tkachenko in 2000.     

Uniqueness of Polish group topology

We show the limits of Mackey's theorem applied to identity sets to prove that a given group has a unique Polish group topology.Verbal sets in Abelian Polish groups and full verbal sets in the infinite symmetric group are Borel. However this is not true in general.A Polish group with a neighborhood π-base at 1 of sets from the σ-algebra of identity and verbal sets has a unique Polish group topology. It follows that compact, connected, simple Lie groups, as well as finitely generated profinite groups, have a unique Polish group topology.     

Structure of geometric wrap groups of non-archimedean fibers

This article is devoted to the investigation of structure of wrap groups of fiber bundles over ultra-normed infinite fields and more generally over Cayley–Dickson algebras. Iterated wrap groups are studied as well. Their smashed products are constructed and studied.     

Compactifications of the homeomorphism group of a graph

Let Γ be a countable locally finite graph and let H(Γ) and H₊(Γ) denote the homeomorphism group of Γ with the compact-open topology and its identity component. These groups can be embedded into the space of all closed sets of Γ×Γ with the Fell topology, which is compact. Taking closure, we have natural compactifications and . In this paper, we completely determine the topological type of the pair and give a necessary and sufficient condition for this pair to be a (Q,s)-manifold. The pair is also considered for simple examples, and in particular, we find that has homotopy type of RP³. In this investigation we point out a certain inaccuracy in Sakai-Uehara's preceding results on for finite graphs Γ.     

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.     

Products of monotonically normal spaces with factors defined by topological games

Recently, it has been proved that orthocompactness implies normality for the products of a monotonically normal space and a compact space. It had been known that normality, collectionwise normality and the shrinking property are equivalent for the same products. We extend these two results for the products replacing the compact factor with a factor defined by topological games. Moreover, we prove the equivalence of orthocompactness and weak suborthocompactness in these products.     

A characterization of inverse limits of Q-bundle maps

In this paper we prove that a surjection between metric compacta is a hereditary shape equivalence if and only if it is an inverse limit of trivial Q-bundle maps. This result was conjectured by T.B. Rushing. Near-homeomorphisms are instrumental to the proof.     

Cauty?s space enhanced

The title refers to Cauty?s example (Cauty, 1994 [3]) of a metric vector space which is not an absolute retract. It is shown that Cauty?s space can be refined to the effect that the completion of the refined space can be isomorphically embedded as a subspace of an F-space which itself is an absolute retract.     

Spaces of rational maps and the Stone-Weierstrass theorem

It is shown that Segal's theorem on the spaces of rational maps from CP¹ to CPⁿ can be extended to the spaces of continuous rational maps from CP^m to CPⁿ for any m?n. The tools are the Stone-Weierstrass theorem and Vassiliev's machinery of simplicial resolutions.     

Precompact abelian groups and topological annihilators

For a compact Hausdorff abelian group K and its subgroup H≤K, one defines the g-closureg_K(H) of H in K as the subgroup consisting of χ∈K such that χ(a_n)?0 in T=R/Z for every sequence {a_n} in (the Pontryagin dual of K) that converges to 0 in the topology that H induces on . We prove that every countable subgroup of a compact Hausdorff group is g-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every g-closed subgroup of a compact Hausdorff group is realcompact. The techniques developed in the paper are used to construct a close relative of the closure operator g that coincides with the G_δ-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.     

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.     

On subgroups of minimal topological groups

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U₁ is the Urysohn universal metric space of diameter 1, the group Iso(U₁) of all self-isometries of U₁ is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.