Some problems of Arhangelʹskii on diagonal-flexible spaces and rotoids

In this short article, we give complete answers to four problems of Arhangelʹskii on diagonal-flexible spaces or rotoids. We list the four problems as follows: m A. Is every compact diagonal-flexible space rectifiable? m B. Is every diagonal-flexible space with a countable π-character first-countable? m C. Is every diagonal-flexible compact space with a countable π-character first-countable? m D. Is every paratopological group a rotoid?