Ryzard Engelking in his topology text at the historical and bibliographic notes of $5.1$ says that a topological space $X$ is fully normal if every open cover has an open star refinement (click here to see original text) but at $3.1$ he defines implicitly (is it true?) refinements must be a cover (click here to see original text) so that more explicitly for Engelking a topological space would be fully normal if every open cover has an open-star refinement which is even a cover. Moreover, even Steen and Seebach in the text Counterexamples in Topology seem to proceed just as Engelking: see here for details. However, on this Wikipedia article is simply written that a topological space is fully normal when every open cover has an open star refinement and is stated that fully normal spaces are normal.
So on this WikiProof page it is proved that an Engelking fully normal space is normal and this proof requires that the open star refinement must be a cover so that I thought to put a question where I ask to prove or disprove if a Wiki fully normal space is normal. So could someone help me, please?
