Show that any basic open set about a point on the "top edge," that is, a point of form [tex](a, 1)[/tex], where [tex]a < 1[/tex], must intersect the "bottom edge."(adsbygoogle = window.adsbygoogle || []).push({});

Background:

Definition-Thelexicographic squareis the set [tex]X = [0,1] \times [0,1][/tex] with the dictionary, or lexicographic, order. That is [tex](a, b) < (c, d)[/tex] if and only if either [tex]a < b[/tex], or [tex]a = b[/tex] and [tex]c < d[/tex]. This is a linear order on [tex]X[/tex], and the example we seek is [tex]X[/tex] with the order topology.

We follow usual customs for intervals, so that [tex][(a,b),(c,d)) = \{ (x,y) \in X : (a,b) \leq (x,y) < (c,d) \}[/tex]. A subbase for the order topology on [tex]X[/tex] is the collection of all sets of form [tex][(0,0),(a,b))[/tex] or of form [tex][(a,b),(1,1)).[/tex]

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# Lexicographic Square, topology

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