Lexicographic Square, topology
Show that any basic open set about a point on the "top edge," that is, a point of form , where , must intersect the "bottom edge."
Definition- The lexicographic square is the set with the dictionary, or lexicographic, order. That is if and only if either , or and . This is a linear order on , and the example we seek is with the order topology.
We follow usual customs for intervals, so that . A subbase for the order topology on is the collection of all sets of form or of form