Chains of pseudocompact group topologies Original Research Article

We show that for several large classes of groups G, the poset %plane1D;4AB;σ(G) of all pseudocompact group topologies of weight σ on G contains a poset isomorphic to the power set of σ whenever %plane1D;4AB;σ(G)≠Ø. This permits to describe in purely set-theoretic terms when G has bounded (from above) chains of pseudocompact group topologies of weight σ. Moreover, we show that these topologies may additionally have some of the following properties: linear (in particular, zero-dimensional), connected and locally connected, disconnected and locally connected, connected and nonlocally connected, etc. It turns out that the question of whether the cardinality of unbounded chains of pseudocompact group topologies of weight ω1 on real is larger than that of bounded ones cannot be answered in ZFC. We answer also questions posed by [11] and [12] on pseudocompact topologization of a given weight.