regular element

A regular element of a Heyting algebra is an element xx such that ¬¬x=x\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 SS is a regular element of SS as a frame.

Analogously, a regular open set in a topological space XX is a regular element of the frame of open sets of XX; 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 xx is ¬¬x\neg{\neg{x}}; note that this is regular. In fact, any element of the form ¬y\neg{y} is regular.

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