nLab interval cut







A way of defining a dense linear order that behaves like the closed intervals in a dense linear order, which identifies each closed interval [a,b][a,b] with the pair of open intervals (,a),(b,)(-\infty, a), (b, \infty).


Given a DLO DD, An interval cut is a pair (L,U)(L,U) of subsets of elements in DD satisfying these conditions:

  1. LL is inhabited; that is, some element of DD belongs to LL.
  2. Similarly, UU is inhabited.
  3. LL is a lower set; that is, if a<ba \lt b are elements of DD with bLb \in L, then aLa \in L.
  4. Similarly, UU is an upper set: if a<ba \lt b are elements of DD with aUa \in U, then bUb \in U.
  5. LL is an upwards open set; that is, if aLa \in L, then a<ba \lt b for some bLb \in L.
  6. Similarly, UU is a downwards open set: if bUb \in U, then a<ba \lt b for some aUa \in U.
  7. If aLa \in L and bUb \in U, then a<ba \lt b.

The DLO of closed intervals in DD is defined to be the set of all interval cuts in DD.


See also


Created on May 6, 2022 at 14:46:01. See the history of this page for a list of all contributions to it.