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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.

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Some obvious ideas:

If $(X,\tau_1)$ is compact, $(X,\tau_1, \tau_2)$ will be $(1,2)$-clopen-compact for any $\tau_2$.

If $\tau_2$ is the trivial topology, then the only $(1,2)$-clopen subsets are $\emptyset$ and $X$, and so the bitopological space is $(1,2)$-clopen-compact. This generalises trivially to the case that $\tau_2$ is finite.

If $\tau_2 \subset \tau_1$ and $(X,\tau_1)$ is clopen-compact (e.g. a connected space, or a compact space..) then $(X,\tau_1, \tau_2)$ is a $(1,2)$-clopen-compact bitopological space. This is because if a set $O$ is $(1,2)$-clopen, it is open in $\tau_1$, closed in $\tau_2$, so closed in $\tau_1$ as well, hence clopen in $\tau_1$.

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Thank you Dear Dr. Brandsma,

It is known that a bitopological space $(X,\tau_1,\tau_2)$ is $p$-connected (here $p$ is breviety of the word "pairwise") iff the class of (1,2)-clopen sets contains only $\emptyset$ and $X$ (the notion of $p$-connectedness introduced by W. Pervin).

In addition to your answer I suggest that all $p$-connected bitopological spaces are $(1,2)$-clopen-compact, too.

But what about nontrivial examples?

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