Dear all who love general topology,
In general topology we know the notion of clopen-compact spaces (introduced by A. Sostak): a topological space $(X,\tau)$ is called clopen-compact if every clopen cover of $X$ admits a finite subcover. Usually, a subset $G$ is called clopen if it is simultaneously open and closed.
Generalizing this for bitopological spaces we get : a subset $B$ of a bitopological space $(X,\tau_1, \tau_2)$ is called (1,2)-clopen if $B$ is open relative to $\tau_1$ and closed relative to $\tau_2$. Note that generally, the class of $(1,2)$-clopen sets does not coincide to the class $(2,1)$-clopen sets.
A bitopological space $(X,\tau_1, \tau_2)$ we call to be $(1,2)$-clopen-compact if $(1,2)$-clopen cover of $X$ admits a finite subcover.
It is interesting to construct example(s) of $(1,2)$-clopen-compact bitopological spaces.