A regular element of a Heyting algebra is an element $x$ such that $\neg{\neg{x}} = x$.

Thus a Boolean algebra is precisely a Heyting algebra in which every element is regular.

As a special case, a regular open in a locale$S$ is a regular element of $S$ as a frame.

Analogously, a regular open set in a topological space$X$ is a regular element of the frame of open sets of $X$; equivalently, an open set which equals the interior of its closure, or equivalently the exterior of its exterior. (This is the origin of the term, related to a regular space.)

The regularisation of $x$ is $\neg{\neg{x}}$; note that this is regular. In fact, any element of the form $\neg{y}$ is regular.

Revised on March 15, 2012 18:06:53
by Urs Schreiber
(82.169.65.155)