- #1
mathsss2
- 38
- 0
Show that any basic open set about a point on the "top edge," that is, a point of form (a, 1), where a < 1, must intersect the "bottom edge."
Background:
Definition- The lexicographic square is the set X = [0,1] \times [0,1] with the dictionary, or lexicographic, order. That is (a, b) < (c, d) if and only if either a < b, or a = b and c < d. This is a linear order on X, and the example we seek is X with the order topology.
We follow usual customs for intervals, so that [(a,b),(c,d)) = \{ (x,y) \in X : (a,b) \leq (x,y) < (c,d) \}. A subbase for the order topology on X is the collection of all sets of form [(0,0),(a,b)) or of form [(a,b),(1,1)).
Background:
Definition- The lexicographic square is the set X = [0,1] \times [0,1] with the dictionary, or lexicographic, order. That is (a, b) < (c, d) if and only if either a < b, or a = b and c < d. This is a linear order on X, and the example we seek is X with the order topology.
We follow usual customs for intervals, so that [(a,b),(c,d)) = \{ (x,y) \in X : (a,b) \leq (x,y) < (c,d) \}. A subbase for the order topology on X is the collection of all sets of form [(0,0),(a,b)) or of form [(a,b),(1,1)).
