Open and Closed Elements
Elements of an interior algebra satisfying the condition xI = x are called open. The complements of open elements are called closed and are characterized by the condition xC = x. An interior of an element is always open and the closure of an element is always closed. Interiors of closed elements are called regular open and closures of open elements are called regular closed. Elements which are both open and closed are called clopen. 0 and 1 are clopen.
An interior algebra is called Boolean if all its elements are open (and hence clopen). Boolean interior algebras can be identified with ordinary Boolean algebras as their interior and closure operators provide no meaningful additional structure. A special case is the class of trivial interior algebras which are the single element interior algebras characterized by the identity 0 = 1.
Read more about this topic: Interior Algebra
Famous quotes containing the words open, closed and/or elements:
“I am Black because I come from the earth’s inside
now take my word for jewel in the open light.”
—Audre Lorde (b. 1934)
“Don: Why are they closed? They’re all closed, every one of them.
Pawnbroker: Sure they are. It’s Yom Kippur.
Don: It’s what?
Pawnbroker: It’s Yom Kippur, a Jewish holiday.
Don: It is? So what about Kelly’s and Gallagher’s?
Pawnbroker: They’re closed, too. We’ve got an agreement. They keep closed on Yom Kippur and we don’t open on St. Patrick’s.”
—Billy Wilder (b. 1906)
“psychologist
It is through friendships that teenagers learn to take responsibility, provide support, and give their loyalty to non- family members. It is also in teenage friendships that young people find confidants with whom to share thoughts and feelings that they are not comfortable sharing with their parents. Such sharing becomes one of the elements of true intimacy, which will be established later.”
—David Elkind (20th century)