Some Baire semitopological groups that are topological groups (Q2405087)

A semitopological group is a group equipped with a topology such that the multiplication is separately continuous. The paper contributes to the study of the well-known problem to find topological conditions under which a semitopological group is a topological group. The main result of the author states that a semitopological group which is a regular \(\Delta\)-Baire space and in which the multiplication is feebly continuous is a topological group. The second part of the paper shows that the class of \(\Delta\)-Baire spaces is surprisingly large. To this end the author makes use of topological games. Therefore the result cited above generalizes many results from the literature. Given a topological space \((X,\tau)\) and following E. Reznichenko, a subset \(W\subseteq X\times X\) is called separately open, in the second variable, if for each \(x\in X,\) \(\{z\in X: (x,z)\in W\}\) belongs to \(\tau\). The space \((X,\tau)\) is said to be a \(\Delta\)-Baire space if for each separately open, in the second variable set \(W,\) containing \(\Delta_X=\{(x,y)\in X\times X:x=y\},\) there exists a nonempty open set \(U\) of \(X\) such that \(U\times U\subseteq \overline{W}^{\tau\times \tau}\).