Open-closed set

A subset of a topological space which is simultaneously open and closed in that space (cf. Open set; Closed set). A topological space $X$ is disconnected if and only if it contains an open-closed set other than $\emptyset$ and $X$. If the family of all open-closed sets of a topological space is a basis of its topology, then this space is called inductively zero-dimensional. Every Boolean algebra is isomorphic to the Boolean algebra of all open-closed sets of an appropriate inductively zero-dimensional Hausdorff compactum. The so-called extremally-disconnected Hausdorff compacta form a special class of zero-dimensional compacta, and are characterized by the fact that the closure of any open set in them is also open (and closed). Every complete Boolean algebra is isomorphic to the Boolean algebra of all open-closed sets of an appropriate extremally-disconnected Hausdorff compactum.

Comments

An open-closed set is also called a closed-open set or clopen set.

The correspondence between Boolean algebras and inductively zero-dimensional compact Hausdorff spaces is known as Stone duality or Stone topological duality. Instead of "inductively zero dimensional" one also finds simply "zero-dimensional topological spacezero dimensional" in the literature.

A zero-dimensional compact Hausdorff space is called a Boolean space.

References

[a1]	S. Koppelberg, "General theory of Boolean algebras" J.D. Monk (ed.) R. Bonnet (ed.) , Handbook of Boolean algebras , 1 , North-Holland (1989) pp. Sect. 3.7