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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{{Cite book|title=Introduction to topological manifolds|author=Lee, John M.|date=2011|publisher=Springer|isbn=978-1441979391|edition=2nd|location=New York|oclc=697506452}}) is a topology on the unit square S, i.e. on the set of points (x,y) in the plane such that {{nowrap|1=0 ≤ x ≤ 1}} and {{nowrap|1=0 ≤ y ≤ 1.}}{{sfnp|Steen|Seebach|1995|p=73}}

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, {{nowrap|1=(x,y) \scriptstyle\prec (u,v)}} if and only if either {{nowrap|1=x < u}} or both {{nowrap|1=x = u}} and {{nowrap|1=y < v}}. Stated symbolically,

(x,y)\prec (u,v)\iff (x

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.{{sfnp|Steen|Seebach|1995|p=66}} 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 for 0\le x\le 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. Its fundamental group is trivial.{{sfnp|Steen|Seebach|1995|p=73}}

See also

Notes

{{reflist}}

References

  • {{Citation|first=L. A.|last=Steen|first2=J. A.|last2=Seebach|title=Counterexamples in Topology|publisher=Dover|year=1995|isbn=0-486-68735-X|title-link=Counterexamples in Topology}}

Category:General topology

Category:Topological spaces