Manuel Sanchis

Product of bounded subsets in paratopological groups.

All spaces under consideration are Tychonoff spaces. A subset B of a space X is said to be bounded in X if the restriction of any continuous real-valued function on X to B is bounded. A space X is called pseudocompact if it is bounded in itself. The two main problems in the context of bounded subsets are related to the product of such subsets. To be precise, if B_\alpha is a bounded subset in X_\alpha (\alpha\in A),

  1. When is the product \prod_{\alpha\in A}B_\alpha\in A bounded in \prod_{\alpha\in A}X_\alpha\in A?
  2. Under what conditions does the equality

    \prod_{\alpha\in A} \text{cl}_{\beta(X_\alpha)}B_\alpha = \text{cl}_{\beta(\prod_{\alpha \in A}X_\alpha)}\prod_{\alpha\in A}B_\alpha

    hold?

Notice that (2) is a version in this context of the Glicksberg’s theorem characterizing when the product of a family of pseudocompact spaces is pseudocompact as well.

It is known that (2) implies (1), and that the converse remains an open problem. Notably, in these types of problems, the properties of the spaces X_\alpha are as crucial as the manner in which the bounded subsets B_\alpha of these spaces are embedded in X_\alpha. For example, within the context of pseudocompact spaces, (1) and (2) are equivalent (Glicksberg’s theorem). When topological and algebraic properties overlap, problems (1) and (2) take on a new perspective: by a theorem of Comfort-Ross, questions (1) and (2) are equivalent for pseudocompact topological groups. Moreover, by a result of Hernández-Sanchis-Tkachenko, the same is valid for bounded subsets in topological groups. In this talk, we will analyze problems (1) and (2) within the framework of paratopological groups. We will examine the main known results, present some new findings related to question (2), and discuss the key open problems in this context.