Consonance and Cantor set-selectors

It is shown that every metrizable consonant space is a Cantor set-selector. Some applications are derived from this fact, also the relationship is discussed in the framework of hyperspaces and Prohorov spaces.peer-reviewe

Some topics in topological algebra

AbstractIn this paper we investigate the topological structure of free topological universal algebras with given continuous signature. We establish some properties of compact subsets of free topological algebras and also some properties of MK-equivalence. Next, we give necessary and sufficient conditions for a free topological algebra to be a k-space, an l-space or a space with countable tightness

Functional equivalence of topological spaces

AbstractLet E be a non-trivial Banach space. The question when the spaces Cp(X,E) and Cp(Y,E) of all continuous mappings of X and Y into E in the topology of pointwise convergence are linearly homeomorphic is studied. These spaces are called lE-equivalent. A topological property or a cardinal function is called lE-invariant if it is preserved by the relation of the lE-equivalence. We prove that σ-discreteness, σ-scatteredness, the hereditary Lindelöf number, the hereditary density, the density, and the spread are lE-invariant properties. Moreover, we prove that in the class of μ-spaces of pointwise countable type the scatteredness, k-scatteredness, the extent, the paracompactness, and the p-paracompactness are lE-invariants. For that we introduce the notions of pm-equivalence, om-equivalence, pom-equivalence. We study some functional functors

Remainders of rectifiable spaces

AbstractWe prove a Dichotomy Theorem: for any Hausdorff compactification bG of an arbitrary rectifiable space G, the remainder bG∖G is either pseudocompact or Lindelöf. This theorem generalizes a similar theorem on topological groups obtained earlier in A.V. Arhangel'skii (2008) [6], but the proof for rectifiable spaces is considerably more involved than in the case of topological groups. It follows that if a remainder of a rectifiable space G is paracompact or Dieudonné complete, then the remainder is Lindelöf and that G is a p-space. We also present an example showing that the Dichotomy Theorem does not extend to all paratopological groups. Some other results are obtained, and some open questions are formulated

Representing spaces as images of zero-dimensional spaces

AbstractThe following result is obtained, and is then applied to give a new proof of a set-valued selection theorem: For every nonempty Tychonoff space Y there exists a Tychonoff space X with dim X = 0, a perfect map f : X → Y, and a paracompact S ⊂ X with dim S = 0, such that f(S) = Y and f ∣ S is open