Abstract:
We note that a pseudocompact space $X$ that was constructed by E. A. Reznichenko is hereditarily metalindelöf. Moreover, every (hereditarily) metalindelöf space $Y$ can be attached to $X$ (the size of $X$ can vary to accommodate $Y$) so that the resulting space is a (hereditarily) metalindelöf pseudocompact space that contains $Y$ as a closed subset. This example is much simpler than related constructions of a pseudocompact not compact space with a point-countable base that are due to S. Watson and
D. B. Shakhmatov or a metalindelöf pseudocompact not compact space that is due to Ian Tree.