Lexicographic order topology on the unit square

In general topology, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square^[1]) is a topology on the unit square S, i.e. on the set of points (x,y) in the plane such that 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.^[2]

Construction

The lexicographical ordering gives a total ordering $\prec$ on the points in the unit square: if (x,y) and (u,v) are two points in the square, (x,y) $\scriptstyle \prec$ (u,v) if and only if either x < u or both x = u and y < v. Stated symbolically, $(x,y)\prec (u,v)\iff (x<u)\lor (x=u\land y<v)$

The lexicographic order topology on the unit square is the order topology induced by this ordering.

Properties

The order topology makes S into a completely normal Hausdorff space.^[3] Since the lexicographical order on S can be proven to be complete, this topology makes S into a compact space. At the same time, S contains an uncountable number of pairwise disjoint open intervals, each homeomorphic to the real line, for example the intervals $U_{x}=\{(x,y):1/4<y<1/2\}$ for $0\leq x\leq 1$ . So S is not separable, since any dense subset has to contain at least one point in each $U_{x}$ . Hence S is not metrizable (since any compact metric space is separable); however, it is first countable. Also, S is connected and locally connected, but not path connected and not locally path connected.^[1] Its fundamental group is trivial.^[2]

Notes

^ ^a ^b Lee, John M. (2011). Introduction to topological manifolds (2nd ed.). New York: Springer. ISBN 978-1441979391. OCLC 697506452.
^ ^a ^b Steen & Seebach (1995), p. 73.
^ Steen & Seebach (1995), p. 66.