Sequential Conditions and Free Topological Groups

Long colimits of topological groups II: Free groups and vector spaces

Weakly complete free topological groups

The topological properties of q-spaces in free topological groups

The topological fundamental group and free topological groups

Notes on countable tightness of the subspaces of free (Abelian) topological groups

S2 and the Fréchet property of free topological groups

We give a complete description of the topological spaces X such that the free abelian topological group A(X) embeds into the free abelian topological group A(I) on the closed unit interval. In particular, the free abelian topological group A(X) on any finite-dimensional compact metrizable space X embeds into A(I). To obtain our description, we study similar embeddings of the free locally convex spaces and continuous surjections between the spaces of continuous functions with the pointwise topology. Proofs are based on the classical Kolmogorov's Superposition Theorem.

The kR-property of free Abelian topological groups and products of sequential fans

It is shown that, under the assumption of the continuum hypothesis, the normality of all finite powers of a topological space does not imply the normality of its free topological group.

A norm on a group G is a function N mapping G into the set of non-negative real numbers such that for each x and y in G, N(xy−1) ⩽ N(x)+N(y) and N(e) = 0, where e is the identity element of G. It is shown here that if F(X) is the free topological group on any completely regular Hausdorff space X and H is a subgroup of F(X) generated by a finite subset of X, then any norm on H can be extended to a continuous norm on F(X).

We prove that the additive group (E*, τ k (E)) of an $${\mathcal {L}_\infty}$$ -Banach space E, with the topology τ k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ k (E)) is uniformly homeomorphic to a subset of $${\ell_2^\kappa}$$ for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.

Показано, что в предположении континуум-гипотезы нормальность всех конечных степеней топологического пространства не влечет нормальности его свободной топологической группы. Библиография: 11 названий.

Fréchet–Urysohn subspaces of free topological groups III

It is shown that the free fc-group on a simply connected locally equiconnected space is simply connected.This result is then used to verify, for a large class of groups, the conjecture of Ordman that itx(G * H) = irx(G) X i7x(H), where G * H is the free product of G and H.

We survey the present trends in theory of universal arrows to forgetful functors from various categories of topological algebra and functional analysis to categories of topology and topological algebra. Among them are free topological groups, free locally convex spaces, free Banach-Lie algebras, and more. An accent is put on the relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and other may become amenable to the methods of this theory.