Connected orderable spaces
Howi Kok · Data Archiving and Networked Services (DANS) · 1973
versity of Amsterdam for the stimulating discussions and the encouragement during the course of the investigations.I also wish to thank Drs. A.E.Brouwer of the Mathematical Centre in Amsterdam for the fact that he participated in the research for some time.A considerable part of the results we obtained together is contained in this tract. CONTENTSA subset J of X is called an open interval if J is of the form J = (a,b) or J = (a, ) or J = ( ,b) or J = X.J is called a closed interval if J is of the form J = [a,b] or J = [a, ) orIf [a,b] = {a,b} where a and bare distinct points of X, then we call a and b neighbours in X; a is the left neighbour of band bis the right neighbour of a.The set {a,b} is called a jump.A pair (A,B) of subsets of an ordered set (X,<) is called a cut, if X =Au B, An B = 0, A# 0, B # 0 and if a< b for all a€ A, b € B. A gap of a totally ordered set (X,<) is a cut (A,B) of X, such that A has no largest element and B has no smallest element.A totally ordered set (X,<) is called order-complete if each non-void subset of X which is bounded above has a supremum in X.It is clear that an ordered set (X,<) is order-complete iff each non-void subset which is bounded below has an infimum in X.Moreover, (X,<) is order-complete if and only if there are no gaps.