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."

**Background:**

**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