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

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The lexicographical ordering gives a total ordering  ≺ {\displaystyle \prec } on the points in the unit square: if (x,y) and (u,v) are two points in the square, (x,y)  ≺ {\displaystyle \scriptstyle \prec } (u,v) if and only if either x < u or both x = u and y < v. Stated symbolically,  ( x , y ) ≺ ( u , v ) ⟺ ( x < u ) ∨ ( x = u ∧ y < v ) {\displaystyle (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

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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 } {\displaystyle U_{x}=\\{(x,y):1/4<y<1/2\\}} for  0 ≤ x ≤ 1 {\displaystyle 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 {\displaystyle 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]

## See also

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* List of topologies
* Long line

## Notes

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

## References

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* Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X