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
editThe lexicographical ordering gives a total ordering on the points in the unit square: if (x,y) and (u,v) are two points in the square, (x,y) (u,v) if and only if either x < u or both x = u and y < v. Stated symbolically,
The lexicographic order topology on the unit square is the order topology induced by this ordering.
Properties
editThe 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 for . So S is not separable, since any dense subset has to contain at least one point in each . 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
editNotes
edit- ^ 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.
References
edit- Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X