On some properties of semi-compact and \(S\)-closed bitopological spaces (Q2709537)

A subset \(A\) of a bitopological space \((X,\tau_1,\tau_2)\) is called \((i,j)\)-semi-open if \(A\subset\tau_j\)-\(\text{Cl}(\tau_i\)-\(\text{Int}(A))\). The bitopological space \((X,\tau_1, \tau_2)\) is said to be extremally disconnected if for every \(S\in \tau_i\) the set \(\tau_j\)-\(\text{Cl}(S)\in \tau_i\). In this paper the notions of semi-compact and \(S\)-closed topological spaces for the bitopological case are generalized. A bitopological space \((X,\tau_1, \tau_2)\) is said to be \((i,j)\)-semi-compact if every of its \((i,j)\)-semi-open covers admits a finite subcover. A bitopological space is said to be \((i,j)\)-\(S\)-closed (resp. \((i,j)\)-quasi-\(H\)-closed) [\textit{M. N. Mukherjee} and \textit{S. Raychaudhuri}, Indian J. Pure Appl. Math. 26, No. 5, 433-439 (1995; Zbl 0846.54015)] if every of its \((i,j)\)-semi-open (resp. \(\tau_i\)-open) covers \(\{U_a:a\in A\}\) contains a finite family \(\{U_{a_k}: 1\leq k\leq n\}\) such that \(X= \bigcup^{k=n}_{k=1} \tau_j\)-\(\text{Cl}(U_{a_k})\). The characterizing properties of \((i,j)\)-\(S\)-closed bitopological spaces in connection with \((i,j)\) semi-compact, \((i,j)\)-quasi-\(H\)-closed and \(P\)-extremally disconnected spaces, where \(i,j\in \{1,2\}\), \(i\neq j\), are given.