Arkady G. Leiderman

The Free Abelian Topological Group and the Free Locally Convex Space on the Unit Interval

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) of the closed unit interval. In particular, the free abelian topological group A(X) of any finite-dimensional compact metrizable space X embeds into A(I). The situation turns out to be somewhat different for free locally convex spaces. Some results for the spaces of continuous functions with the pointwise topology are also obtained. Proofs are based on the classical Kolmogorovs Superposition Theorem.

Uniform Eberlein compactifications of metrizable spaces

Abstract We prove that each metrizable space X (of size | X | ⩽ c ) has a (first countable) uniform Eberlein compactification and each scattered metrizable space has a scattered hereditarily paracompact compactification. Each compact scattered hereditarily paracompact space is uniform Eberlein and belongs to the smallest class A of compact spaces, which contains the empty set, the singleton, and is closed under producing the Alexandroff compactification of the topological sum of a family of compacta from the class A .

On linear continuous open surjections of the spaces CP(X)

Solving a problem by Arkhangelskiĭ, we construct a linear continuous open surjection L : Cp(X) → CP(Y) for compacta X and Y with 0 < dim X < dim Y < ∞. An example of nonopen linear continuous surjection of the space Cp[0, 1] onto itself is given. Related topics and results are discussed.