nLab
regular element

This entry is about the concept in formal logic and topology. For the concept in physics/quantum field theory see at regularization (physics).

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. These define the regular open sublocales of SS. We may also say regular open subspace for this (or the following concept).

An analogous kind of regular open subspace is a regular open set in a topological space XX, which is a regular element of the frame of open sets of XX. Equivalently, this is an open set in XX that 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 regularization of xx is ¬¬x\neg{\neg{x}}; note that this is regular. In fact, any element of the form ¬y\neg{y} is regular. Note that x¬¬xx \leq \neg{\neg{x}}; in logic, this means that a double negation is a weaker statement.

In a topological space, the regularization of an open set GG can be constructed as Int(Cl(G))Int(Cl(G)), or equivalently as Ext(Ext(G))Ext(Ext(G)). Sometimes one performs this operation to an arbitrary set (in the space) to produce a regular open set. But note that while GInt(Cl(G))G \subseteq Int(Cl(G)) when GG is open, this does not hold for an arbitrary set.

Last revised on December 11, 2019 at 12:13:30. See the history of this page for a list of all contributions to it.